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peclassified 

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memo 2 August 196® 


SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 



This document contains information affecting the national defense of the 
United States within the meaning of the Espionage Act, 50 U. S. C., 
31 and 32, as amended. Its transmission or the revelation of its contents 
in any manner to an unauthorized person is prohibited by law. 

This volume is classified^ K^^-n K in accordance with security 
regulations of the War and Navy D ej^^tmen^ because certain chapters 
contain material which wag at the date of printing. 
Other chapters may have had a lower classification or none. The reader 
is advised to consult the War and Navy agencies listed on the reverse of 
this page for the current classification of any material. 


J^F(I)E^TJAL 


Manuscript and illustrations for this volume were pre- 
pared for publication by the Summary Reports Group of the 
Columbia University Division of War Research under con- 
tract OEMsr-1131 with the Office of Scientific Research and 
Development. This volume was printed and bound by the 
Columbia University Press. 

Distribution of the Summary Technical Report of NDRC 
has been made by the War and Navy Departments. In- 
quiries concerning the availability and distribution of the 
Summary Technical Report volumes and microfilmed and 
other reference material should be addressed to the War 
Department Library, Room lA-522, The Pentagon, Wash- 
ington 25, D. C., or to the Office of Naval Research, Navy 
Department, Attention: Reports and Documents Section, 
Washington 25, D. C. 


Copy No. 


This volume, like the seventy others of the Summary Tech- 
nical Report of NDRC, has been written, edited, and printed 
under great pressure. Inevitably there are errors which 
have slipped past Division readers and proofreaders. There 
may be errors of fact not known at time of printing. The 
author has not been able to follow through his writing to 
the final page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports 
and sent to recipients of the volume. Your help will make 
this book more useful to other readers and will be of great 
value in preparing any revisions. 



SUMMARY TECHNICAL REPORT OF DIVISION 7, NDRC 


VOLUME 1 


GUNFIRE CONTROL 



By 


SEP 1 

« memo 2 August I960 


pefense ttierao 

UBBABX 01 




OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

DIVISION 7 
H. L. HAZEN, CHIEF 



WASHINGTON, D. C., 1946 


NATIONAL DEFENSE RESEARCH COMMITTEE 


James B. Conant, Chairman 
Richard C. Tolman, Vice Chairman 
Roger Adams Army Representative^ 

Frank B. Jewett Navy Representative- 

Karl T. Compton Commissioner of Patents^ 

Irvin Stewart, Executive Secretary 


^ Army representatives in 07'der of sei'vice: 

Maj. Gen. G. V. Strong Col. L. A. Denson 

Maj. Gen. R. C. Moore Col. P. R. Faynionville 

Maj. Gen. C. C. Williams Brig. Gen. E. A. Regnier 

Brig. Gen. W. A. Wood, Jr. Col. M. M. Irvine 

Col. E. A. Routheau 


-Navy representatives in order of service: 

Rear Adm. H. G. Bowen Rear Adm. J. A. Furer 
Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 
Commodore H. A. Schade 
3 Co7nmissio7iers of Patents in order of service: 
Conway P. Coe Casper W. Ooms 


NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Committee 
were (1) to recommend to the Director of OSRD suitable 
projects and research programs on the instrumentalities 
of warfare, together with contract facilities for carrying 
out these projects and programs, and (2) to administer 
the technical and scientific work of the contracts. More 
specifically, NDRC functioned by initiating research 
projects on requests from the Army or the Navy, or on 
requests from an allied government transmitted through 
the Liaison Office of OSRD, or on its own considered ini- 
tiative as a result of the experience of its members. Pro- 
posals prepared by the Division, Panel, or Committee for 
research contracts for performance of the work involved 
in such projects were first reviewed by NDRC, and if 
approved, recommended to the Director of OSRD. Upon 
approval of a proposal by the Director, a contract per- 
mitting maximum flexibility of scientific effort was ar- 
ranged. The business aspects of the contract, including 
such matters as materials, clearances, vouchers, patents, 
priorities, legal matters, and administration of patent 
matters were handled by the Executive Secretary of 
OSRD. 

Originally NDRC administered its work through five 
divisions, each headed by one of the NDRC members. 
These were: 

Division A — Armor and Ordnance 
Division B — Bombs, Fuels, Gases, & Chemical Prob- 
lems 

Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three 
administrative divisions, panels, or committees were cre- 
ated, each with a chief selected on the basis of his out- 
standing work in the particular field. The NDRC mem- 
bers then became a reviewing and advisory group to the 
Director of OSRD. The final organization was as follows : 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Fire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 — Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 


iv 


CONFIDE \TIAL 


3 


library of Congress 



460885 


NDRC FOREWORD 


As EVENTS of the years preceding 1940 re- 
vealed more and more clearly the serious- 
ness of the world situation, many scientists in 
this country came to realize the need of or- 
ganizing scientific research for service in a 
national emergency. Recommendations which 
they made to the White House were given care- 
ful and sympathetic attention, and as a result 
the National Defense Research Committee 
[NDRC] was formed by Executive Order of 
the President in the summer of 1940. The mem- 
bers of NDRC, appointed by the President, 
were instructed to supplement the work of the 
Army and the Navy in the development of the 
instrumentalities of war. A year later, upon 
the establishment of the Office of Scientific Re- 
search and Development [OSRD], NDRC be- 
came one of its units. 

The Summary Technical Report of NDRC is 
a conscientious eifort on the part of NDRC to 
summarize and evaluate its work and to pre- 
sent it in a useful and permanent form. It 
comprises some seventy volumes broken into 
groups corresponding to the NDRC Divisions, 
Panels, and Committees. 

The Summary Technical Report of each Di- 
vision, Panel, or Committee is an integral sur- 
vey of the work of that group. The first volume 
of each group’s report contains a summary of 
the report, stating the problems presented and 
the philosophy of attacking them, and sum- 
marizing the results of the research, develop- 
ment, and training activities undertaken. Some 
volumes may be “state of the art” treatises 
covering subjects to which various research 
groups have contributed information. Others 
may contain descriptions of devices developed 
in the laboratories. A master index of all these 
divisional, panel, and committee reports which 
together constitute the Summary Technical Re- 
port of NDRC is contained in a separate vol- 
ume, which also includes the index of a micro- 
film record of pertinent technical laboratory 
reports and reference material. 

Some of the NDRC-sponsored researches 
which had been declassified by the end of 1945 
were of sufficient popular interest that it was 
found desirable to report them in the form of 
monographs, such as the series on radar by 
Division 14 and the monograph on sampling 
inspection by the Applied Mathematics Panel. 
Since the material treated in them is not dupli- 
cated in the Summary Technical Report of 
NDRC, the monographs are an important part 
of the story of these aspects of NDRC research. 


In contrast to the information on radar, 
which is of widespread interest and much of 
which is released to the public, the research on 
subsurface warfare is largely classified and is 
of general interest to a more restricted group. 
As a consequence, the report of Division 6 is 
found almost entirely in its Summary Tech- 
nical Report, which runs to over twenty vol- 
umes. The extent of the work of a Division can- 
not therefore be judged solely by the number 
of volumes devoted to it in the Summary Tech- 
nical Report of NDRC ; account must be taken 
of the monographs and available reports 
published elsewhere. 

The Fire Control Division, initially Section 
D2 under the leadership of Warren Weaver and 
later Division 7 under Harold L. Hazen, made 
significant contributions to an already highly 
developed art. It marked the entrance of the 
civilian scientist into what had hitherto been 
regarded as a military specialty. 

It was one of the tasks of the Division to 
explore and solve the intricate problems of 
control of fire against the modern military air- 
craft. Gunnery against high-speed aircraft in- 
volves fire control in three dimensions. The 
need for lightning action and superlatively 
accurate results makes mere human skills hope- 
lessly inadequate. The Division’s answer was 
the development of the electronic M9 director 
which, controlling the fire of the Army’s heavy 
AA guns, proved its worth in the defense of 
the Anzio Beachhead and in the protection of 
London and Antwerp against the Nazi V- 
weapons. In addition to producing mechanisms 
such as the M9, the Division made less tan- 
gible but equally significant contributions 
through the application of research methods 
which had a profound, even revolutionary, in- 
fiuence on fire control theory and practice. 

The results of the work of Division 7, for- 
merly Section D2, are told in its Summary 
Technical Report, which has been prepared at 
the direction of the Division Chief and has 
been authorized by him for publication. It is 
a record of creativeness and devotion on the 
part of men to whom their country will always 
be grateful. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. CONANT, Chairman 
National Defense Research Committee 


CONFIDENTIAL 


v 




FOREWORD 


is THE termination of the war approached, 
x^the members of Division 7, most of whom 
had left active professional work to engage in 
full-time work for Division 7, were subjected 
to strong pressure to pick up their normal 
peace-time activities. Their obligations were so 
heavy that the task of getting out a Summary 
Technical Report was formidable for a number 
of the areas of Division 7 activities. That this 
first volume has actually been produced is due 
in large measure to the continued effort of Mr. 
L. M. McKenzie, the Division's editor for its 
Summary Technical Report and the person who 


has done a major part of the work of prepar- 
ing Volume 1. His competence as Technical Aide 
in the Washington office of the Division inevi- 
tably brought him in close touch with much of 
the Division’s activity. He therefore had an ex- 
cellent background for taking over the responsi- 
bility of assembling this volume. For doing this 
task under difficult circumstances and doing it 
well, the members of Division 7, and particularly 
its Chief, are greatly indebted to Mr. McKenzie. 

H. L. Hazen 
Chief, Division 7 



vii 



PREFACE 


rpHE SUMMARY TECHNICAL REPORT of Division 
-L 7, NDRC, comprises three volumes. They 
differ considerably from one another in method 
of presentation as well as in subject matter. 
The present volume briefly summarizes the 
work of Sections 7.1, 7.3, 7.5, and 7.6 of Di- 
vision 7, and also lists contract numbers, ser- 
vice control numbers, personnel, and other 
factual data relating to the Division. Volume 
2, dealing with optical range Anders, reports 
in detail the work of Section 7.4. Volume 3 
gives an extensive account of the activities of 
Section 7.2 on airborne Are control systems. 

Throughout Part I of the present volume the 
reader is merely introduced to the researches 
and developments sponsored by a Section. He 
is expected to consult the references for details 
about a particular subject. This should not be 
objectionable, for the Division has through- 
out its life issued Reports to the Services sum- 
marizing in some detail the work in a particu- 
lar field or upon a particular device. These have 
been widely distributed and are, with various 
other reports, included in the microfilmed 
record. 

Part II of this volume, written by Dr. R. B. 
Blackman, Dr. H. W. Bode, and Dr. C. E. 
Shannon of the Bell Telephone Laboratories, 
departs from this pattern. It is a detailed tech- 
nical treatise on smoothing of data and repre- 
sents material which, if written during the war, 


would have been issued as a report to the Ser- 
vices. It is included here since it highlights a 
particular contribution to the fire control art 
which was of especial importance in the activi- 
ties of the Division. 

In keeping with the general character of the 
present volume. Dr. Harold L. Hazen, Chief of 
Division 7, has written the opening chapter. 
This bears upon all three volumes by orienting 
the reader with respect to the activities of the 
Fire Control Division. With the future worker 
in mind, he highlights those aspects of the 
Division 7 program which are particularly 
significant. 

Finally, a word with regard to the inevitable 
errors and inconsistencies which may be pres- 
ent in the remainder of the text. Although 
compiled principally from the writings of other 
technical aides and members of the Division, 
certain interpretations and explanations may 
not faithfully reflect original meanings, and the 
writer assumes the responsibility for any dis- 
tortion. In particular, Duncan J. Stewart and 
George R. Stibitz contributed to Chapter 2, E. 
J. Poitras to Chapters 3 and 4, Warren Weaver 
and George R. Stibitz to Chapter 5, and Ivan 
A. Getting to Chapter 6. 

L. M. McKenzie 
Technical Aide, Division 7 


ix 




CONTENTS 


PART I 

GUNFIRE CONTROL 

CHAPTER PAGE 

1 Fire-Control Activities of Division 7, NDRC . 3 

2 Land-Based Fire Control 12 

3 Servomechanisms 38 

4 Pneumatic Controls 44 

5 Mathematical Analysis of Fire-Control Prob- 
lems 54 

6 Seaborne Fire Control with Radar .... 62 

PART 11 

DATA SMOOTHING AND PREDICTION 

IN FIRE-CONTROL SYSTEMS . . 71 

7 General Formulation of the Data-Smoothing 

Problem 75 

8 Steady-State Analysis of Data Smoothing . . 85 

9 The Assumption of Analytic Arcs .... 100 

10 Smoothing Functions for Constants . . . 107 

11 Smoothing Functions for General Polynomial 

Expressions 112 

12 Physical Realization of Data-Smoothing Func- 
tions 117 

13 Illustrative Designs and Performance Analysis 125 

14 Variable and Nonlinear Circuits 134 

Appendix A 145 

Appendix B 156 

Bibliography 161 

OSRD Appointees 168 

Contract Numbers 170 

Service Project Numbers 174 

Index 177 







PART I 

GUNFIRE CONTROL 





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Chapter 1 


FIRE-CONTROL ACTIVITIES OF DIVISION 7, NDRC 

By H. L. Hazen 


U NDER THE URGENCY of war, Division 7 of 
NDRC, in common with other development 
agencies, stressed useful results rather than 
integrated summary records. Reports there 
were in great numbers, many of them excellent 
and valuable, but the broad objectives and 
overall plan for the division’s work were car- 
ried largely in men’s minds. Each of the divi- 
sion members knew which projects were im- 
provisations against time, which were aimed 
at fundamental gains, and which were specula- 
tive ventures with promising but unproven 
ideas. If the enduring values contained in the 
many hundreds of man-years’ work of able 
scientists and engineers invested in the divi- 
sion’s program are to be of continuing use, 
they must be reasonably accessible to the 
future worker. To provide this accessibility, an 
overall view of the program is needed by means 
of which the future worker can see from the 
division’s point of view the objectives, nature, 
and significance of the various projects, and 
know to what sources to go to find the pertinent 
results. It is the purpose of this Summary 
Technical Report to supply this overall view 
and to point out these sources. 

To accomplish this purpose best within the 
limitations of manpower available for prepar- 
ing this report, it has been necessary to permit 
a treatment that is nonuniform in style and 
approach. In some areas the work has been 
previously reported very fully in the form of 
Reports to the Services so that a brief state- 
ment for orientation and introduction to the 
bibliography suffices. In other areas the indi- 
vidual reports have treated specialized details 
so that here a comprehensive logical exposition 
seems necessary. Thus this summary is designed 
to complement previous reports in such a way 
that the two taken together form a consistent 
whole. 

This chapter serves as an introduction to the 
three volumes of the Division 7 Summary 
Technical Report. It is an attempt to high light 
and appraise in such a way that the reader 


may know what the author in retrospect 
regards as the more important subjects for 
future attention. First, however, certain ob- 
servations concerning the nature of fire control 
and its development prior to World War II 
will perhaps serve to establish common ground 
even though they may be a statement of the 
obvious. 

FIRE CONTROL 

^ ^ ^ Scope of Fire Control 

Fire control is here understood to include 
means of directing missiles of all sorts in such 
a way as to cause them to strike desired 
targets. Thus it is associated with all forms of 
guns, with rockets, with bombs dropped from 
airplanes whether they be free-falling or 
guided, with torpedoes, with depth charges 
dropped from ships or airplanes, to mention 
some of the more important categories. While 
the work of the division was associated with 
all of these areas, the intensity of effort varied 
widely and in a number of areas other divi- 
sions involved in missile development did the 
major part of the fire-control work for their 
missiles. 

Fire control has reached its highest stage 
of evolution thus far in the heavy antiaircraft 
artillery field. There are excellent reasons why 
this is so. First, antiaircraft action is so 
rapid, the problem is so complicated, and the 
results are required to be so accurate that 
simple human skill is utterly useless. Very high 
quality instrumentation alone has any chance 
of success. In contrast, field artillery, for exam- 
ple, achieves high precision but its targets are 
essentially stationary and the time for work- 
ing up fundamental data is relatively great. 
Furthermore, the field artillery problem is 
largely two-dimensional in character. The 
Navy main-battery surface-fire problem may 
involve moving targets and is complicated by 
a moving gun platform, but here the problem 


3 


4 


FIRE-CONTROL ACTIVITIES OF DIVISION 7, NDRC 


is still two-dimensional and the time scale is 
relatively expanded in comparison with that 
for antiaircraft fire. 

Control of various forms of fire from air- 
craft is an equally difficult problem in its tech- 
nical elements but the stringent weight and 
space limitations on airborne equipment pre- 
clude the elaborateness and refinement permis- 
sible in the heavy surface and antiaircraft fire 
control. 

Dynamical Unity of a Fire-Control 
System 

Fire control involves very complex physical 
apparatus, among the most complex that has 
been devised. Because of this, as well as of the 
security aspect mentioned subsequently, the 
theory of dynamical behavior was in a very 
primitive state prior to World War II. This 
has meant that there frequently has been dif- 
ficulty in distinguishing malfunctioning due to 
mechanical imperfection or design from that 
due to violation of dynamical laws. One aspect 
of this, the essential dynamical unity of a 
complete fire-control system is so important as 
to justify the following remarks. 

Functionally, fire-control apparatus may be 
roughly separated into three components. The 
first is the data-gathering elements which 
provide information concerning the target 
position and motion, own-ship’s orientation, 
position, and velocity, and secondarily wind, 
air density, and other correction data. The 
second component performs the computation 
functions. It determines from all the data fur- 
nished by the first component how the missile 
shall be launched in order to arrive at some 
given point in space at the same instant that 
the target is at that point. The third com- 
ponent, not always thought of as strictly fire 
control, includes the data-transmission and 
power elements by which the data calculated 
by the computer are caused to control the gun, 
the automatic pilot or other means of launch- 
ing the missile. 

The separation of the three components, for 
purposes of analysis and understanding, is 
justifiable on much the same basis that one 
separates in a living organism for purposes of 


study the circulatory, respiratory, and nutri- 
tive functions. One must always remember, 
however, that a fire-control system, just as a 
living organism, is more than the sum of com- 
ponent parts. It is an integrated whole with 
interrelated functioning of all its parts and 
one is safe in considering the parts separately 
only if one always keeps in mind their relation 
to the whole. This is a fundamental fact of ut- 
most importance. 

113 Future Use of Fire Control 

One may, and in fact should, ask wherein 
the fire control of World War II is pertinent 
to the methods of warfare we shall have to use 
in the future if mankind continues so primitive 
as to find fighting the only way of settling 
disputes. The future warfare appears to place 
great emphasis on long-range missiles of great 
destructive power guided largely automatically 
to their targets — perhaps automatically seek- 
ing them. This war’s fire control is pertinent 
because even if the specific fire-control appa- 
ratus used in connection with this war’s ord- 
nance were not applicable, the fundamental 
principles of fire control for guided missiles 
are precisely the same as those underlying the 
fire control of World War II. Component 
mechanisms and techniques are essentially the 
same. Furthermore, the concept of dynamical 
unity is even more comprehensive in its scope 
in the guided missile problem, for here it must 
include guiding and homing functions, in addi- 
tion to the three component functions of con- 
ventional fire control. Therefore, the fire con- 
trol of the future will be merely an extension 
and outgrowth of fundamental principles and 
techniques of today. For this reason it appears 
worth while to attempt to make the results of 
the substantial efforts in fire control during 
World War II accessible to future workers. A 
second, but not a negligible, reason why this 
war’s fire control is pertinent, is that history 
seems to indicate that old and seemingly 
primitive techniques often retain their value 
in war for a surprisingly long time after newer 
methods have made their appearance. That is, 
we are far from sure that essentially conven- 
tional antiaircraft artillery, conventional 


PREWAR STATUS OF FIRE CONTROL 


5 


bombing by airplanes, and conventional plane- 
to-plane gunfire are obsoleted by newer devel- 
opments. 

1-2 PREWAR STATUS OF 

FIRE CONTROL 

The role that Section D-2 and Division 7 
played in World War II can perhaps be better 
understood in terms of the prewar status of 
lire control. Viewed as a technical field, there 
are two outstanding characteristics of fire con- 
trol. First, it is a highly technical subject re- 
quiring an extraordinarily high level of com- 
petence in the research, development, and pro- 
duction phases on the part of professional 
scientists, engineers, and craftsmen. Second, it 
is by nature one of the more highly classified 
and isolated areas of military endeavor. The 
latter is naturally so because of the very great 
importance of fire control in securing domina- 
tion of an enemy and because the almost in- 
spired type of creativeness required makes the 
preservation of security highly important on 
the part of those who have achieved high com- 
petence.. 

Near the beginning of World War II, the 
United States was favored by having a few 
groups who under a tight veil of secrecy had 
shown ingenuity and skill amounting practi- 
cally to genius in the development and pro- 
duction of fire-control gear. Their mechanical 
design and craftsmanship show a courage, a 
persistence in the face of formidable difficulties, 
a mastery of complication, and a refinement in 
execution that command the highest respect. 
However, the isolation inevitably bred by secu- 
rity necessarily cut off these groups not only 
from association with scientists and engineers 
at large, but even from each other. These are 
naturally not the conditions that yield the best 
that science and technology can produce. How- 
ever, it did produce surprisingly good results 
in this country even with the severe restric- 
tions that were imposed. 

When Section D-2 entered the scene, it found 
this highly developed art characterized by skill 
and craftsmanship of superlative grade, partic- 
ularly in mechanical design and construction. 
It was an accomplishment in which any group 


working under these conditions could justifiably 
take great pride. Anyone entering the field 
fresh even with an extraordinarily strong 
scientific background would necessarily find 
himself faced with months of hard study 
merely to catch up with what had been done. 
The field of fire control had been worked over 
intensively and with great ingenuity within 
the scope of techniques possessed by the small 
quota of individuals who were given access to 
and permitted to work in this field. The thing 
that new groups broadly grounded in science 
might be expected to bring to such a field are 
the benefits of a new and fresh point of view 
and of a range and breadth of experience over 
a variety of fields that could see relations be- 
tween fire control and many varied fields of 
endeavor that, superficially viewed, are un- 
related to it. 

A good example of the benefit of a fresh 
point of view is the development that led to 
the antiaircraft director M9 for the Army’s 
heavy antiaircraft guns. This development 
represented two significant departures from 
previous work. The first is the bringing to bear 
of communications techniques on the problem 
of smoothing and prediction. The second is the 
development of electrical computing techniques 
to replace the previous mechanical techniques. 

Superficially there is no apparent relation 
between fire control and electrical communica- 
tions. More fundamentally, however, both are 
concerned with the separation of useful in- 
formation or data from the unwanted but una- 
voidable data in the form of “noise” or rough 
tracking. In fact, ultimate performance of 
equipment in both fields is limited fundamen- 
tally by the extent to which these two, the 
wanted information and the unwanted or 
spurious information, can be separated. The 
adaptation of the methods already highly 
developed for this purpose in the communica- 
tions field into a form useful in the fire-control 
field constitutes one of the important contribu- 
tions of World War II to fire control. For this 
reason, the comprehensive and fundamental 
monograph on data smoothing and prediction* 
is included as Part II of Volume 1. 

^ By R. B. Blackman, H. W. Bode, and C. E. Shannon 
of the Bell Telephone Laboratories. 


6 


FIRE-CONTROL ACTIVITIES OF DIVISION 7, NDRC 


There are numerous other examples of the 
benefits obtained by cross-fertilization between 
fields. It is believed that the personnel of 
Section D-2 and Division 7, representing as 
they did rather diverse backgrounds coupled 
in each case with a broad scientific training, 
were able to bring about substantial contribu- 
tions in many areas of fire control. 

13 DIVISION 7 CONTRIBUTIONS 

Developments Put in Service 

One may ask what specific contributions 
Division 7 made to the winning of World War 
II. There are a few concrete items and, in 
addition, a large number of intangibles whose 
contribution it is impossible to assess with 
accuracy. 

In the tangible category the antiaircraft 
director M9 which formed so successful a 
partner with the SCR-584 radar and the prox- 
imity fuse undoubtedly played an important 
role on a number of occasions among which 
may be mentioned the antiaircraft defense at 
the Anzio Beachhead and the V-1 defense of 
England and Antwerp. Section D-2 played a 
major role in the development of this director 
by the Bell Telephone Laboratories. 

Two other Division 7 developments that 
reached the active theaters in substantial quan- 
tities are the oil-gear power drive M3B1 and 
the M7 sighting system both for the 37-mm 
and 40-mm guns. The first of these was devel- 
oped at the Massachusetts Institute of Tech- 
nology and produced by Westinghouse. The 
second was developed by Pitney-Bowes. The 
oil-gear drive was apparently very satisfactory, 
since it went into the field in large numbers 
with practically no resulting complaints. The 
M7 sighting system (the course-and-speed 
sight often known as the Weissight) produced 
varying responses as did the other competitive 
fire-control systems used with the Bofers gun. 
None of these was ideal. 

1-3* Fundamental Performance Data 

A second and somewhat less tangible con- 
tribution of Division 7, though probably no 
less real, was the influence exerted by in- 


creasing emphasis on the vital importance of 
meaningful quantitative data on fire-control 
system performance. Prior to World War II 
no equipment existed for testing fire-control 
systems or even important components under 
dynamical reproducible conditions. This em- 
phasis was reinforced and implemented by a 
series of developments of testing machines. 
Conspicuous among these are the Barber-Col- 
man and the Bell Telephone Laboratories 
dynamic testers for antiaircraft fire control; 
the Texas tester for plane-to-plane fire control ; 
and the Patuxent River plane-to-plane fire- 
control testing establishment set up for the 
Navy. These and various less imposing similar 
developments, together with the division’s 
perpetual insistence upon significant appraisal, 
undoubtedly exerted major influence on fire- 
control thinking both in the United States and 
abroad. 

Consulting Activity 

A third major but quite intangible contribu- 
tion is the influence exerted by essentially con- 
sulting activities with many branches .of both 
Services by able men associated with the divi- 
sion. Scientific and technical counsel from 
division personnel were sought by many 
Service groups on innumerable occasions prior 
to the making of important decisions. The 
overall effect of such consultation of the war 
effort is impossible to appraise. It is believed 
to have been substantial. 

1 ^ SUGGESTIONS FOR FUTURE WORK 

After these preliminary observations, we 
turn now, in the remainder of this chapter, to 
a rapid review of some of those aspects of the 
division’s work to which it is believed the 
future worker can with profit give some atten- 
tion. To avoid undue complexity and detail 
which might obscure the main points, only the 
more important items are mentioned. 

1. Fundamental theory. 

2. Testing and appraisal. 

3. New principles and components. 

4. Matching mechanisms to men. 

5. Optical range finders. 


CONFIDENTIAL 


SUGGESTIONS FOR FUTURE WORK 


7 


For the reader who desires further informa- 
tion more detail is given in subsequent chapters 
of this Summary Technical Report. From this, 
in turn, the reader can go to the original 
reports. 

' ^ Fundamental Theory 

One of the more important studies in this 
field is that mentioned previously, namely the 
monograph on data smoothing and prediction 
(Part II of this volume). Another fundamental 
study of the prediction process'’ is summarized 
in Chapter 5 of this volume. These are proba- 
bly the two most important efforts of the divi- 
sion in fire-control theory. This work has had 
and will continue to have a fundamental in- 
fluence on subsequent thinking. 

From a somewhat different point of view 
fire-control theory has been influenced sub- 
stantially by statistical studies such as are 
reviewed in Chapter 5. The measure of the 
effectiveness of ordnance is the probability of 
damage to a target which depends on many 
factors including operating procedures and 
tactics as well as fire control. Studies showing 
how each factor affects the probability of 
damage to target are therefore of great im- 
portance as a guide to development showing 
where effort should be placed to secure the 
greatest gains. Much work was done in this 
field, including some by Section 7.5 of Division 
7, as summarized in Chapter 5. 

In the airborne fire-control field the sub- 
stantial contributions to fundamental theory^ 
are treated in a unified manner in Part I of 
Volume 3. This treatment embraces nearly all 
phases of airborne fire control including gun- 
fire, bombing, rocketry, and aerial torpedoing, 
and represents much effort and analysis. In- 
cidentally, the brief section on torpedo fire-con- 
trol theory,'^ Part II of Volume 3, should not 
be overlooked as a possibility for capitalizing 
on the high-grade “present-range” information 
available by radar. 

Under the subject of fundamental theory 
should be noted the several publications spon- 

By Norbert Wiener. 

c By George A. Philbrick. 

^ By A. L. Ruiz. 


sored by Division 7 on the theory of servo- 
mechanisms. These have been widely circulated 
in response to many requests. 

Other contributions to the fundamental 
theory of fire control are found scattered 
throughout the division’s reports but the fore- 
going indicates the more comprehensive con- 
tributions. 

Testing and Appraisal 

As has been stated, the division considers 
that one of its major contributions is the 
strong and persistent emphasis on meaningful 
quantitative data as the only sound basis for 
appraisal of the technical performance of fire- 
control equipment. The two principal areas in 
which this feeling was influential are in the 
heavy antiaircraft problem and the problem of 
plane-to-plane fire control. Relatively late in 
World War II preliminary studies were made 
for adequate instrumentation in the automatic 
weapons gunnery field and in the bombing field. 
In the latter two, however, the adequate instru- 
mentation appeared to require a longer term 
of development than was justified by the stage 
the war had reached. The bombing analyzer 
study was undertaken preliminarily by the 
Eastman Kodak Co. The automatic weapons 
accessory problem was given exploratory study 
by the Armour Research Foundation. 

Returning to testing in the antiaircraft ar- 
tillery field. Section D-2 early became im- 
pressed with both the exceeding costliness and 
the inadequacy of existing testing and ap- 
praisal methods. These were limited primarily 
to the quite unsatisfactory towed-target field 
tests and to component tests consisting largely 
of static measurements. The prediction process 
fundamentally requires elements whose dy- 
namic performance in response to rapidly 
changing data is essentially different from its 
static performance; consequently the only 
significant tests are those that effectively simu- 
late dynamic conditions. The mechanical dy- 
namic tester developed by the Barber-Colman 
Co. and the two models of tape dynamic tester 
later developed by the Bell Telephone Labo- 
ratories marked very important steps in this 
process of obtaining adequate instrumentation 


CONFlDMinAL 


8 


FIRE-CONTROL ACTIVITIES OF DIVISION 7, >DRC 


for dynamic tests of fire-control equipment 
under reproducible conditions. Implementing 
these and the involved overall trial firing tests 
are a series of computing machines and aids, a 
data recorder, and other auxiliary instruments 
by means of which these processes are speeded 
up and their time and money costs reduced. 
Both the dynamic tester and these computing 
aids are considered to be important tools in 
future developments. 

As pointed out above, however, the testing 
of fire control for automatic weapons is in a 
most unsatisfactory state. Serious attention 
should be given to it in the future. 

In the field of the plane-to-plane fire control 
even more expensive and elaborate develop- 
ments were required than for heavy antiair- 
craft, in order to provide even a good first ap- 
proximation to satisfactory testing and assess- 
ment techniques. Two major developments of 
the division stand out as of continuing sig- 
nificance, namely, the Texas tester and the 
Patuxent River testing project. 


I. 4.3 ]Vew Principles and Components 

Out of the numerous projects carried on by 
the division came many new ideas or modifi- 
cations of old ideas which did not achieve 
practical embodiment by the end of World War 

II. A number of these ideas, however, are 
believed to be worth the attention of future 
workers in the field and are pointed out here 
for that purpose. 

In the field of antiaircraft fire control (under 
Section 7.1 of Division 7) which is covered in 
Chapter 2 of the present volume, particular 
attention is called to the following projects. 
The report of the Bryant Chuck Grinder Co. 
contains a substantial number of novel sug- 
gestions for fire-control mechanisms, some 
quite unconventional, that deserve careful 
study. While the M9 director proved the value 
of electrical computation techniques in fire 
control, it by no means exhausted the possi- 
bilities of electrical computation; in fact it 
serves merely as an introduction to a field that 
justifies far more extensive exploitation. The 
experimental T15 director contains ideas 


worthy of merit as do directors involving elec- 
trical techniques produced experimentally by a 
number of groups under direct contracts with 
the Services and having no connection with 
NDRC. Certainly further attention in the 
heavy antiaircraft fire-control field should be 
given to other than M9 electrical techniques as 
well as to possible new mechanical computer 
techniques and, probably more important, vari- 
ous combinations of electrical and mechanical 
means of computation. 

The prediction of future position of targets 
fiying accelerated paths is one which has arisen 
in the past and will undoubtedly arise fre- 
quently in the future. On this subject the ex- 
tensive series of investigations undertaken by 
the Bell Telephone Laboratories under Division 
7 and Ordnance Department auspices are 
worthy of attention. 

In the automatic weapons field, but having 
implications for heavier guns, are the T28 
director developed by Eastman Kodak Co. and 
the course-invariant sight developed by the 
Baker Manufacturing Co. The T28 embodies 
a direct representation of three-dimensional 
vector quantities that seems inherently adapted 
to a problem that is fundamentally three-di- 
mensional in character. 

The Baker course-invariant sight also in- 
volves novel principles having certain inherent 
advantages that appear to justify further 
development. 

In the naval field, the gunfire-control system 
Mark 56 is believed to represent a significant 
advance in the fire control of heavy Navy and 
antiaircraft guns. This set is going into limited 
production so that appraisal under sea con- 
ditions will be possible. It benefited from the 
major advances in fire-control thinking 
achieved during World War II. 

Turning now to airborne fire control, there 
is an extended series of devices developed at 
the Franklin Institute and elsewhere for gun- 
nery, bombing, torpedoing, and rocket firing 
that near the end of the war were foiming into 
a single pilot’s universal sighting system 
[PUSS]. These are covered in Volume 3, 
Chapter 10. This work may be of value for two 
reasons. First, the successful ideas should be 
continued, and second, there is an extensive 


SUGGESTIONS FOR FUTURE WORK 


9 


series of negative results which can be re- 
viewed with benefit by anyone seriously in- 
terested in this field. 

In the plane-to-plane gunnery field, the work 
of the Franklin Institute applied primarily to 
fixed gun installations. In the field of turret 
gunnery, the developments at the General Elec- 
tric Co. on airborne computers contain numer- 
ous ideas of interest in the future development 
of airboime central-station computers. 

In calling attention to new principles and 
components that may be of interest in future 
work, some emphasis can well be placed on the 
extensive exploration and partial exploitation 
of pneumatic techniques by Section 7.3 of Divi- 
sion 7, NDRC. The low-level bombsights, Mark 
23 and Mark 25, represent the fire-control uses 
of pneumatic methods for computation and 
energy amplification. In the field of lead-com- 
puting sights for either surface or possibly 
aerial use, the pneumatic methods partially in- 
vestigated under the Eastman Kodak Co. 
project are worth further attention. It is be- 
lieved that in various computation and servo 
applications, particularly where a fast response 
and light weight are at a premium, pneumatic 
methods are worth very careful examination 
and substantial further development. These 
matters are discussed in Chapter 4. 

Section 7.3 also sponsored a considerable 
amount of work on servomechanisms other 
than the pneumatic type. These are discussed 
in Chapter 3. The gun drives for the Army 37- 
and 40-mm guns, for example, achieved notable 
success. 


Matching Mechanisms to Men 

Growing out of the major activity on the 
subject of optical range finders was a sub- 
stantial amount of work on the combined per- 
formance of operators and machines. In par- 
ticular, many investigations were aimed at 
determining what properties or parameters of 
machines, such as tracking devices, for exam- 
ple, enabled the man to produce most precise 
results with the machine. Of these studies the 
extensive work on tracking done by the Fox- 
boro Company (under Section 7.4, reported in 


Volume 2) and the Franklin Institute (under 
Section 7.2, reported in Volume 3) deserve 
particular attention from persons interested in 
the manual tracking process. These studies 
have produced results in such form as to be in 
many cases immediately applicable for desig- 
ners of manually controlled fire-control equip- 
ment. The techniques employed are also sug- 
gestive for those who wish to pursue this sub- 
ject farther. 


® Optical Range Finders 

While by the end of World War II the spec- 
tacular success of radar methods of determin- 
ing target position had demonstrated their ac- 
curacy, reliability, and resistance to counter- 
measures (when intelligently designed and 
used), it was not until nearly then that these 
qualities of radar had sufficiently demonstrated 
themselves to obviate intensive work on optical 
range finders. Range finder work was started 
very early by the division because preliminary 
examination of the fire-control problem had 
clearly indicated that the largest instrumental 
contributor to errors in antiaircraft firing was 
the optical range finder. While much of this 
work will be of interest only if optical range 
finders should for some unforeseen reason 
again assume major importance in warfare, 
there are certain results that have future in- 
terest. One of these is the Harvard University 
studies on the fundamental mechanisms of 
visual distance determination. The results there 
obtained have a profound significance in rela- 
tion to any future range finder development 
because they extend our knowledge of the fun- 
damental distance perception process into 
entirely new territory. 

The benefits of another major cooperative 
enterprise among the Army, the Navy, the 
three major range finder manufacturers, and 
Division 7 have already been passed on to the 
Services who took over the Division 7 contracts 
with the three range finder manufacturers. 
This so-called “super range finder” project 
aimed to embody all the results previously ob- 
tained in a series of designs for superior per- 
fonnance. This work had not felt the impact 


CONFIDENTIAL 
- 


7 


10 


FIRE-CONTROL ACTIVITIES OF DIVISION 7, NDRC 


of the Harvard work, which indeed calls for 
rather revolutionary changes in optical range 
finders if the highest possible precision is to 
be obtained. 

The work associated with optical range 
finders under Division 7 is tremendous in bulk, 
but Volume 2 of this report® should enable the 
worker interested in the subject to gain access 
to pertinent material in this field with relative 
ease. 

15 CONCLUDING REMARKS 

In conclusion a few words about the method 
of operation of Division 7 may be of interest. 
Almost complete autonomy was granted to the 
various divisions of NDRC in their method of 
operation, with the result that each division 
largely developed its own pattern for work. 
Division 7 consisted of a group of professional 
scientific people who, with few exceptions, 
devoted full time to the scientific and technical 
direction of the Division 7 activity. In addition 
they carried the necessary administrative re- 
sponsibility as far as policy matters were con- 
cerned. This group, assisted by technical aides 
and frequently by contractors’ personnel, col- 
lected information, originated ideas, formu- 
lated programs, and supervised projects in 
some detail. The project supervision was 
mainly exercised by mutual discussion of the 
problem with contractors’ personnel, with a 
joint mapping of the future course. In other 
words, the division personnel served in an 
advisory and co-worker status rather than as a 
giver of orders. The latter was necessary only 
as a legality to formalize conclusions previously 
arrived at by mutual discussion and under- 
standing. It can be said with entire honesty 
that the division-contractor relations under 
this arrangement were, with few if any excep- 
tions, characterized by complete cordiality, 
frankness, and complete absence of personal 
conflict or serious friction. 

Division meetings in general were held once 
a month. They were usually for one day 
although not infrequently a two-day meeting 
was held at some center of interest, with one 
day devoted to business, and the second to re- 

e By S. W. Fernberger. 


viewing the work of some project or contractor. 
Characteristically, the meetings were very free, 
uninhibited discussions of projects, technical 
problems and policies, carried on between men 
who were good personal friends but who were 
at the same time strong individualists with 
well-considered convictions. Most conclusions 
were reached only after thorough discussion, 
which occasional visitors were sure involved 
personal rancor until they learned that Divi- 
sion 7 business was done in this way without 
acrimony. These sessions were very stimulat- 
ing. 

The various sections within the division each 
had followed their own individual patterns of 
activity. Several, namely Sections 7.1, 7.3, and 
7.4, functioned informally. Their work was 
done at frequent but irregular meetings of two 
or three persons. Section 7.2, on the other hand, 
operated in much the same way as the division 
itself. It ordinarily held its meetings the day 
before the division meeting with the result 
that its recommendations to the division were 
usually clearly formulated and in a mature 
state when they came up for division discussion 
and action. Section 7.6, which was only begin- 
ning to acquire vigor when the first false 
OSRD demobilization wave struck, also had 
formal meetings at which technical matters 
were discussed and programs formulated. 
Section 7.5 was an administrative convenience 
without expectation of regular section activity. 

All in all the accomplishments of Division 7 
are believed to be substantial. They are not so 
spectacular or so revolutionary as those of 
some of the other divisions. This may be due 
to the personnel or their method of operation. 
It is believed that at least one other factor had 
a rather prominent influence, however, and 
that is the relative maturity of the fire-control 
field as compared with some other fields, such 
as microwave radar, where almost completely 
new techniques were created. 

In any case it is the judgment of the author 
of this chapter, who joined the division after 
its pattern had in many ways been well estab- 
lished, that whatever the results the division 
achieved, its activity was characterized by 
wholehearted immersion in and absorption by 
the work to be done, by no little imagination. 


i <S ^FIDEN'1IA 


CONCLUDING REMARKS 


11 


by a high order of technical intelligence, and 
by at least a reasonable degree of success as 
judged by its overall record. As pointed out 
earlier in this chapter, its greatest influence 
and effectiveness was probably in its consulting 
services rather than in the apparatus devel- 
oped. 


The writer of this chapter is permitted to 
make the foregoing statements because he 
joined the division rather late and therefore 
can qualify with respect to most of the divi- 
sion’s activity rather as an observer than as 
one who took a primary part in the solution of 
technical problems. 


LNFIDENTIAL 


LAND-BASED FIRE CONTROL 


2 1 GENERAL ORGANIZATION OF 
ACTIVITIES OF SECTION 7.1 

^ Evolution of Section 7.1 

Responsibilities 

R eference to a pre-Pearl Harbor history 
of Section D-2 of the National Defense Re- 
search Committee [NDRC] shows the Fire- 
Control Committee as initially committed to 
investigation and development of land-based 
antiaircraft fire-control methods and means" to 
the exclusion of airborne and seaborne fire- 
control devices. 

Indeed, during the early days of Section 
D-2, studies were made of the fire-control field 
generally; but it was clear that the Army 
Ground Forces offered the greater opportunity 
for service and accomplishment. This con- 
clusion was derived partially from a technical 
appraisal of the situation, but even more from 
the fact that the Army Ground Forces were 
less satisfied than the Navy or the Army Air 
Forces with the quality of its own equipment 
and more disposed to welcome cooperative ef- 
forts of Section D-2 in carrying out new de- 
velopments. With the passing of some months. 
Section D-2 (later Division 7) undertook heavy 
programs in contribution to airborne and sea- 
borne fire control, and also continued to ex- 
pand its efforts in the field of land-based fire 
control. The termination of hostilities found 
Division 7 still deeply committed to this work 
under the auspices of Section 7.1. 

The first program of Section D-2 contained 
two broad classes of work: (1) a study of fun- 
damental problems, and (2) a theoretical and 
experimental attack on the specific development 
of a more effective fire-control system for 
heavy antiaircraft weapons. A completely or- 
derly and systematic execution of the section 
program should have proceeded by first study- 
ing fundamental problems and following this 
by developing the necessary instruments. But 
the exigencies of the situation required the 


less ideal and more expensive approach of car- 
rying out both aspects of the program simul- 
taneously and in more or less opportunistic 
fashion, and the section undertook an immedi- 
ate program of antiaircraft director develop- 
ment. 

These early studies quickly required Section 
D-2 to broaden its program, and two further 
classes of work were undertaken. The first 
of these sprung from a lack of adequate instru- 
ments and procedures for the measurements, 
analyses, and assessments of the performance 
of fire-control instruments and components. 
The second was an extension of effort into the 
research and development of fire-control means 
for automatic weapons. 

With the reorganization of Section D-2 as 
Division 7, Section 7.1 was made responsible 
for these classes of Section D-2 projects and 
two additional categories were to appear: (1) 
fire-control instruments for other weapons of 
a miscellaneous character, but pertaining to 
the land-based fire-control problem, and (2) 
certain short-base range finders which were 
begun under Section D-2 and Division 7 aus- 
pices, but which pertained directly to the land- 
based fire-control problem. 


^ Classification of Projects 

For the purposes of a summary technical 
report of Section 7.1 activities, the projects 
which were the responsibilities of that section 
will be treated in this chapter under the fol- 
lowing six categories: 

1. Major caliber antiaircraft weapons and 
gun directors. 

2. Fundamental studies. 

3. Automatic weapons. 

4. Miscellaneous weapons. 

5. Short-base range finders. 

6. A testing program. 

As indicated already, categories 1 and 2 are 
phases of a single problem and, although seg- 


12 


teONFIDEN4I. 


MAJOR CALIBER ANTIAIRCRAFT WEAPONS 


13 


regated, they should be considered as mutu- 
ally supporting. This report attempts to sum- 
marize the nature of the problems involved 
under the various contracts and to describe 
the progress made. For details of these de- 
velopments the reader will be referred to ap- 
propriate documents. 

2 2 MAJOR CALIBER ANTIAIRCRAFT 
WEAPONS (GUN DIRECTORS) 

^ ^ ^ Electrical Gun Director M9 
(Project 2) 

Perhaps one of the most outstanding proj- 
ects of Section D-2, if not the most outstand- 
ing, was the development of the electrical gun 
director M9 by the Bell Telephone Labora- 
tories. (Project 2, contract NDCrc-127.) 

This project was undertaken primarily in 
the hope that it would be possible to produce 
an electrical design which would present two 
advantages over the standard mechanical de- 
sign (the Sperry M7 director) : first, that it 
would be readily procured in large numbers, 
precision electrical equipment as a general rule 
being more easily fabricated than precision 
mechanical equipment, and second, that a di- 
rector of improved performance would result. 
The second objective turned out to be of great 
importance, although at the time the project 
was undertaken, the limitations of and diffi- 
culties with the standard Army mechanical 
director M7 were not appreciated, particularly 
for use with radar. As it turned out, the first 
objective was not accomplished, for the new 
director was more expensive than the old. 

In the model designated TIO/ and in the 
resulting standard director M9, all conversions 
to rectangular coordinates, prediction, ballistic 
computations, and reconversions to polar co- 
ordinates are electrical. Present azimuth and 
elevation are normally obtained by manual 
operation (but with aided tracking) of a track- 
ing head on which the operators ride (Figure 
1). However, the necessary selsyns are pro- 
vided so that, through manual matching of 
pointers, the tracking head may be driven from 
remotely received values of present azimuth 
and elevation. Movement of the tracking head 


operates potentiometers to insert voltages 
which are a function of the azimuth and ele- 
vation as inputs to the computer. Either slant 
range or height can be received from an Ml 



Figure 1. TIO director; the tracker. 


height finder in a separate unit, and used with 
a manual follow-up to operate another poten- 
tiometer to supply a voltage as the range input 
to the computer. Radar range is also accepted 
directly from a potentiometer mounted on the 
range unit. 

The prediction is theoretically exact and the 
sensitivity of individual amplifiers and poten- 
tiometers was held to approximately one-tenth 
of one per cent of full scale. All the computa- 
tions, including many of those for the ballis- 
tics, depend upon the use of rather large, 
highly accurate, nonlinear potentiometers. 

The ballistic computer (Figure 2) is very 
elaborate, being adjustable for wind, drift, 
parallax, muzzle velocity, and dead time. Large 
numbers of amplifiers and potentiometers are 
used in these computations in addition to those 
involved in the prediction. The outputs are 
converted to mechanical motions which drive 
the usual selsyn transmitters. 

TIO was completed in November 1941, and 
a series of tests was carried on at the Anti- 


';CONFIDENTIAL 


57 


14 


LAND-BASED FIRE CONTROL 



aircraft Artillery Board at Fort Monroe, Vir- 
ginia, immediately afterwards.^ Although the 
results were not significantly better than those 
obtained from the standard Sperry director, 


Figure 2. TIO director; view of computer show- 
ing amplifier frame detached for maintenance 
work. 

without wind, drift, muzzle velocity, and other 
nonstandard conditions, analysis of the results 
indicated that the errors could be greatly re- 
duced, perhaps to one-half their then existing 
value, by introducing improved smoothing cir- 
cuits, the theory for which was developed 
under Projects 6 and 11 to be reviewed below. 

Despite the unfinished nature of the experi- 
ment and the rather poor results, standardiza- 
tion was recommended, but at the same time 
an extension of the NDRC contract was pro- 
vided to incorporate the new smoothing circuit 
and other changes in TIO. This combination 
was tested and the expected improvement was 
realized. The modified director was also tested 
with radar XT-1 (later designated SCR-584) 
and the combination found to give very satis- 
factory results. 


The M9 director embodied the improvements 
made in TIO and many others and, with SCR- 
584, gave an outstanding performance during 
World War II. 


Gun Director T15 (Project 30) 

During the development of the first electri- 
cal director under Project 2, it became appar- 
ent that there were many alternative electrical 
methods for solving the different problems 
which arose. The exploration of the alterna- 
tive methods suggested a radically different 
approach than that used in the M9 electrical 
director and the development of a director. 


Figure 3. T15 front view of electrical and 

mechanical bays with bay covers removed. 

designated T15,^ was begun under contract 
NDCrc-127 with the Bell Telephone Labora- 
tories, but most of the actual progress and 
all of the construction and testing were car- 
ried out under a new contract, OEMsr-353 


MAJOR CALIBER ANTIAIRCRAFT WEAPONS 


15 


(Project 30) with the Bell Telephone Labora- 
tories, established especially for that purpose. 

While director TIO (standardized as M9) 
was largely electrical in design, T15 was an 
attempt to combine electrical and mechanical 
techniques (Figure 3) using each of these 
techniques in such a way as to produce an 
optimum overall design. T15 utilized the so- 
called “one-plus” feature involving the addi- 
tion of the prediction vector and ballistic cor- 
rections by means of mechanical differentials 
to the present-position vector to give gun 
orders. Prediction is a linear extrapolation to 
the future position along a straight line join- 
ing some selected past position or “memory 
point” to the present position. This feature 
eliminates the rate measuring and smoothing 
problems which are inherent in the design of 
TIO. Alternating rather than direct current 
was used throughout. 

Tests were made on T15 at Camp Davis be- 
ginning in December 1942. Although its per- 
formance was an improvement over that of 
M9, the difference in the two directors was 
masked for many types of courses by larger 
errors outside of the control of the director. 
Full-scale production of M9 was already well 
under way and the improvements in T15 were 
not considered to be sufficiently significant for 
standardization. 


® Curved Flight Computers 

Subsequently, director T15 was modified to 
include fairly accurate prediction against cer- 
tain types of curved courses (see Section 
2.7.3). This modification was designated the 
TlS-El.** The T15-E1 (Figure 4) was based 
on the assumption that the pilot of an enemy 
aircraft would move his controls relatively in- 
frequently. This modified director took ac- 
count of the interchange of potential and 
kinetic energies of the airplane in a diving 
or climbing course. In theory, the T15-E1 
would predict perfectly if the pilot of the craft 
maintained a constant climb or glide angle 
and held all his controls in a fixed position, 
and provided that the absolute motion of the 
plane in space, which is the sum of its motion 


with respect to the air mass, plus the motion 
of the air mass with respect to ground, shall 
be that which can be accommodated by the 
computer. 



Figure 4. AA director T15-E1, showing control 
panel of curved flight bay with cover removed. 
The “joystick” controls rate of turn and angle 
of dive. Throttle and drag controls are to left of 
joystick. Meters in upper view are used for con- 
trol of curved flight predictor. Meters in lower 
row are used for supervision of “memory point” 
predictor. 

This director was compared in a series of 
tests with two other types of curved course 
directors: one a modification of M9 to be dis 
cussed below (Project 78), and another, in- 
volving a plotting board, developed by the Ord- 
nance Department. The plotting board, desig- 
nated the T12, consisted of a surface on which 
pens traced the present and predicted positions 
of the target, assuming a constant accelera- 
tion which was introduced manually; an ob- 
server watching the two pens attempted to 



16 


LAND-BASED FIRE CONTROL 


make the future-position pen predict the course 
of the target by adjusting the acceleration in- 
puts. 

Development of Second-Derivative 
Curvature Attachment for M9 
(Project 78) 

The other attempt to include curvature in 
the prediction of an antiaircraft director re- 
ferred to above is represented by a different 
and perhaps simpler solution to the curvature 



Figure 5. Front view of second-derivative com- 
puter. 

problem and is designed as a plug-in modifi- 
cation of director M9. This work was done 
at the Bell Telephone Laboratories under con- 
tract OEMsr-1263 (Figure 5). 

The Bell Telephone Laboratories suggested 


that the linear prediction mechanism could be 
modified by the introduction of factors propor- 
tional to the second derivatives of the rectan- 
gular coordinates of the target course. The 
M9 was thus adapted so that it would measure 
the acceleration of the target and predict along 
the path in which the acceleration in each rec- 
tangular coordinate remained fixed in both di- 
rection and magnitude. One rather obvious 
disadvantage with this arrangement is that a 
constant angular acceleration in the heading of 
the target does not result in constant accelera- 
tion in the rectangular coordinates. Under 
Project 11 to be reported in Section 2.3.3, the 
theoretical considerations involved in the 
smoothing problem and in the accelerations in- 
volved in the second-derivative application 
were thoroughly investigated. Also, a typical 
but smooth target course was laid out to large 
scale and a prediction based on these data was 
made with the help of a relay interpolator to 
be reported in Chapter 5 (Project 70). This 
was done first with linear prediction, then with 
second-derivative, and finally with third-deriva- 
tive prediction and the results were compared. 
The scores for the three types of prediction in- 
dicated that second-derivative curved flight 
prediction would improve antiaircraft Are. The 
reader is referred to the final technical report 
under the contract for further details.® 


Comparative Testing 

All three curved flight directors were tested 
at the Bell Telephone Laboratories® by means 
of courses (over 200) on a tape dynamic tester 
with tapes cut to represent several measured 
airplane courses, as well as theoretical courses. 
The courses generated by the dynamic tester 
and the future positions predicted by the three 
computers were recorded by the T12 plotting 
board. Further tests were run at the Anti- 
aircraft Artillery Board at Fort Bliss. It was 
concluded that while curved flight prediction 
offered some hope of improved antiaircraft 
performance, the disturbance of the prediction 
by errors in the tracking seriously limited the 
improvement. It is possible that better smooth- 
ing means and more accurate radar data would 


' pOXFIDENTIA|j)v 


MAJOR CALIBER ANTIAIRCRAFT WEAPONS 


17 


improve the performance of curved flight pre- 
dictors, but the termination of World War II 
halted any developments in that direction. 

2 . 2.6 Modification of M7 Director for 
Field Conversion (Project 51) 

As a result of experience gained in the analy- 
sis and design of smoothing circuits in con- 
nection with the development of the electrical 
directors at the Bell Telephone Laboratories, 
new and much more effective types of circuits 
were devised for smoothing the fluctuations 
of input signal to an antiaircraft predictor. 

Contract OEMsr-791 was accordingly negoti- 



Figure 6. Side view of M7 director with produc- 
tion version of smoothing" attachment rate 
matching mechanism (K5-9051). 


ated with the Bell Telephone Laboratories with 
the object of designing a piece of auxiliary 
equipment to be installed on the standard M7 
director (Figure 6), embodying this smooth- 
ing technique applied to the rectangular co- 
ordinate rates. The smoothed rates were in- 
serted in the M7 director through the wind 
mechanism.^ In order to do this, a mechanical 
clutching arrangement had to be inserted in 

^XFII 


the shafts which supply the prediction multi- 
pliers so that the only rate input to these mul- 
tipliers would be the one supplied through the 
wind dials; i.e., the smoothed rates. 

A parallel development was carried on by 
the Ordnance Department at the United Shoe 
Machinery Corp. Both developments were con- 
sidered satisfactory for standardization, and 
a recommendation to that effect was made by 
the Antiaircraft Artillery Board. At that time 
it was NDRC’s understanding that this device 
was being put into production, but it appears 
that none was produced. 

^ ^ ^ Smoothing Device for Employment 
with Radio Set SCR-268 (Project 64) 

Under Ordnance Directive OD-122, contract 
OEMsr-899 was let with the Bristol Company 
(Project 64) to develop a device by means of 
which SCR-268 could be used more effectively 
with an M7 director (Figure 7A).® SCR-268 
supplies radar data on the position of a target 
but gives rather poor values with large and 
violent fluctuations. M7, on the other hand, 
does not perform well unless it is supplied with 
fairly smooth data, since its amplification fac- 
tor is high for frequencies of the order of those 
in the fluctuations. Furthermore, the present- 
position errors of SCR-268 are large. 

Present azimuth, present elevation, and pres- 
ent height, which are received from the SCR- 
268, are resolved into rectangular coordinates 
and plotted. The apparatus includes a set of 
three charts (Figure 7B) on which the rectan- 
gular coordinates of the target are plotted 
automatically as a function of time. The opera- 
tor matches a line to the best average of the 
plot, giving, through a synthesizer, smooth 
values of Eq, Hq, H, x, and y. The first three 
are transmitted to the M7 director and the 
others indicated at the plotter. Only the first 
three can be used unless the director has been 
modified so that rectangular rates can be set 
in by hand. Directors used in the United States 
were not so modified, although NDRC under- 
took a development (and the Ordnance Depart- 
ment undertook a parallel development) for so 
modifying the directors. The NDRC develop- 
ment was under Project 51, reported below. 



18 


LAND-BASED FIRE CONTROL 



Figure 7. A. Side view of T1 smoother, with covers removed. B. Top view of T1 smoother show- 
ing wax-coated charts on which are plotted each rectangular component of target position as func- 
tion of time (chart motion). 





FUNDAMENTAL STUDIES 


19 


Means to correct the parallax between SCR- 
268 and M7 is also provided. The smoother was 
delivered to Camp Davis and was quite effective 
and useful for its original purpose; i.e., to 
smooth SCR-268 data for use with M7. By the 
time the testing was completed SCR-268's were 
being rapidly replaced by SCR-584’s, the out- 
put of which was accurate and quite steady. 
Trials of the smoother with SCR-584 and M7 
showed that the results with and without the 
smoother were about the same, and as a result 
no further work was done on this item. 

2 . 2.8 Plotting Board T9 (Project 64) 

Under Ordnance Directive OD-132, a prob- 
lem was set up as Project 64 with the Bristol 
Company for the development of a semiauto- 
matic plotting board (Figure 8).'’ Inasmuch as 





Figure 8. View of the (“Tenney”) plotting 
board T9. 


this plotting board was not standardized and 
the necessity for a plotting board was probably 
eliminated by the excellent combination of the 
SCR-584 and the M9, the reader is referred to 
the final report under the project for a descrip- 
tion of the apparatus. 


2 3 FUNDAMENTAL STUDIES 

Introduction 


projects under this heading were conceived 
early in NDRC history their lines of activity 
are not so clear as in later projects which fell 
naturally into the organization of work already 
under way. These are among the pioneering 
projects of the Fire-Control Division. In fact, 
the work under Project 17 with the Eastman 
Kodak Co., contributed not only to fundamental 
studies, but to other activities of Section 7.1 
as well, and also to the activities of Sections 
7.2, 7.3, and 7.4. It approached the status of a 
Division 7 central laboratory in fire-control 
matters. 

Project 17, together with Project 11 with the 
Bell Telephone Laboratories, was notably suc- 
cessful. The latter, contributing so materially 
to the Division 7 program as it finally evolved, 
and certain of its results are such an important 
contribution to the development of fire-control 
instruments generally, that a treatise has been 
written covering these aspects of the project, 
and is the subject of Part II of this volume. 

Unlike the other projects of the section, 
those just enumerated did not contribute to 
what might be briefiy called the production of 
hardware. They were set up with the intention 
to explore fields which might lead to devices 
for improving antiaircraft fire control. As 
might be expected, in some cases such explora- 
tion led to the development and design of 
equipment — “hardware’' — and in other in- 
stances results were negative in the sense that 
it did not appear worth while to carry the 
project through to the stage of a finished 
mechanism. The production of improved hard- 
ware is, of course, of vital importance and as 
such is a tangible thing which is relatively 
easy to assess. However, the value of develop- 
ments which produce only additional informa- 
tion, and that sometimes of a negative charac- 
ter, is not always minor. This the reader must 
judge for himself. 


The six projects (Projects 4, 11, 12, 17, 48, Geometrical Predictor Design 

and 68) devoted to this grouping of Section 7.1 at GIT (Project 4) 

activity were initiated prior to the reorganiza- 
tion of Section D-2 as Division 7, and two It was the original expectation of Section 
were terminated prior to that date (December D-2 that the California Institute of Technology 
7, 1942). Since the purposes of the various under contract NDCrc-164 (Project 4) would 


GONFIDEMIAE?; 


20 


LAND-BASED FIRE CONTROL 


conduct a broad survey of mechanical and 
other units capable of carrying out the various 
mathematical operations used in fire-control 
directors. The interest of the group, however, 
led them to work on geometrical-type predic- 
tors. 

The course of any target traveling in a 
straight line, together with the director loca- 
tion, determines a slant plane; and the predic- 
tion problem is a two-dimensional one within 
that plane. If one can determine that plane 
geometrically, lay out the line of flight in that 
plane, and directly extrapolate the distance 
traveled in that plane in proportion to the time 
of flight, the result will be the true future loca- 
tion of the target. 

The California Institute of Technology 
group submitted a predictor design of this 
type, in which the target motion is resolved 
into a slant plane, and the computed lead angle 
in that plane set forward in the main geomet- 
rical representation. Their system determines 
the slant plane by means which are partly 
manual and obtains the speed of the course in 
that plane by means of a speedometer wheel. 

The slant plane system in itself is a good 
approach, but it is not new ; the mechanization 
suggested does not seem to be very practicable, 
and it would be difficult to achieve the neces- 
sary accuracy. 

The project was terminated after submission 
of the preliminary report.^® 


2.3.3 Fundamental Director Studies at the 

Bell Telephone Laboratories (Project 11) 

During the development of the TIO electrical 
gun director by the Bell Telephone Labora- 
tories, it became apparent that there were 
many alternative electrical methods for solving 
the different problems which arose. Contract 
NDCrc-178 was accordingly initiated by the 
section in February 1941, with the Bell Tele- 
phone Laboratories, to study the alternatives 
more fully.^^ 

Reference has already been made above to 
the studies which resulted in the development 
of director T15 and the relatively simple addi- 
tion to the M9 computer as a prediction mech- 


anism for certain types of curved flight. A 
primary outcome of this study was the em- 
phasis it placed upon smoothing in prediction 
theory. This result is deemed of such impor- 
tance in the work of Division 7 that a treatise 
on this subject, which is here published for the 
first time, forms part of the technical report 
under the contract. (Part II of this volume.) 


Analytical Study of Prediction 
Devices and Construction and Tests, 

Iowa State College (Project 12) 

Contract OEMsr-165 (Project 12) was ini- 
tially concerned with prediction theory and 
director design. However, the particular type 
of prediction proposed turned out to be too in- 
accurate. Several other interesting ideas on 
director design were suggested but it was 
decided not to develop these ideas further. 

In addition, this group did considerable work 
on tracking experiments. They constructed 
electrical analogues of mechanical aided track- 
ing circuits. Their apparatus was constructed 
so that straight line courses of different speeds 
and crossover distances can be tracked, using 
different proportions of direct, velocity, and 
acceleration tracking, and using different types 
of controls, such as ordinary handwheels, small 
knobs, and double knobs. The use of small knobs 
was suggested by the hypothesis that perhaps 
more accurate tracking could be attained 
through the use of the fine muscle control of 
the fingers rather than the more gross muscu- 
lature of the arms or back. 

The project was terminated after a report 
of these tracking experiments was received.^^’^* 


2.3.5 Electronic Computing Devices for 
Predictors, RCA (Project 48) 

Under an earlier contract with the Navy, 
the RCA Manufacturing Co. carried out some 
work on the use of electronic computing tech- 
niques for fire-control purposes. The Navy 
asked that NDRC take over this work and 
coordinate it with other projects on electronic 
computers. Conferences were held between the 


^NflDE-MIAL^ 


FUNDAMENTAL STUDIES 


21 


electronic computer groups of RCA, Bell Tele- 
phone Laboratories, Eastman Kodak Co., The 
Massachusetts Institute of Technology, and the 
National Cash Register Co., and as a result 



Figure 9. The computron. 


contract OEMsr-591 was negotiated with RCA 
for this purpose. 

Considerable progress was made in studying 
computing devices in which variables are 
represented by discrete impulses, by unidirec- 
tional voltages or currents, by alternating volt- 
ages or currents (amplitude only) , and by use 
of the phase angle of an alternating voltage or 
current. Experimental units were built to carry 
out the arithmetical operations of addition and 
multiplication, and a unit started for the pur- 
pose of generating a function of two variables. 
Using standard tubes, the computing mech- 
anism becomes so bulky as to be impractical. 
RCA designed a special multiple beam tube, 

called the “computron,”^^ ^g0 Qf which 

reduces the number of tubes needed for com- 
puting by a factor of 100 or more (Figure 9). 
Development of this tube and a computer using 
it was the principal purpose of this contract. 

The division decided not to extend this con- 
tract beyond the date of its termination. It be- 
came clear that there was almost no possibility 
of this development to reach the stage of field 


use in a reasonable time. Furthermore, the 
latest directors had their errors reduced to a 
point where further reduction was of question- 
able value until other factors affecting disper- 
sion could be improved. 

2.3.6 Mechanical Director for 90-mm Anti- 

aircraft Guns, Bryant Chucking 
Grinder Co. (Project 68) 

Development under contract OEMsr-1137 
with the Bryant Chucking Grinder Co. was 
originally contemplated for the purpose of pro- 
ducing a mechanical director for the 90-mm 
batteries which would be less complicated than 
the electrical director T15 (see Sections 2.2.2 
and 2.2.3) but which would have accuracy 
greater than that obtained in previous me- 
chanical directors. Practical considerations 
such as simplicity of operation, ease of main- 
tenance and repair, were to be important con- 
siderations. The so-called "‘one-plus’'*" method 
of handling the lead and the memory point sys- 
tem of obtaining target rates were to be given 
serious consideration. It was also expected that 
curved flight prediction would be included in 
this new mechanical director. 

The Bryant Chucking Grinder Co. undertook 
the project with the understanding that ini- 
tially they would complete only a fundamental 
study of the various problems involved, and 
present recommendations either for or against 
building the director therein contemplated. The 
study was not completed at the termination of 
World War II but the contractor’s final report 
contains many interesting and original ideas 
in connection with the prediction, smoothing, 
and ballistic problems."^ 

2.3.7 Fire-Control Research at the 
Eastman Kodak Co. (Project 17) 

A fundamental proposition always before 
Division 7 was the question of sponsoring a 
central laboratory, which, under direct division 
guidance, would investigate fire-control matters 

The method is called one-plus because the values 
computed are only those to be added to present position 
coordinates and not the total values. 


22 


LAND-BASED FIRE CONTROL 


of diverse nature. Two Division 7 projects ap- 
proached this status, one at the Franklin In- 
stitute under Section 7.2 sponsorship (see 
Volume 3), and another at the Eastman Kodak 
Co. The latter contract was particularly broad 
and concerned work of Sections 7.1, 7.2, 7.3, 
and 7.4. Because of the variety of work done 
at the Eastman Kodak Co. under contract 
OEMsr-56 (Project 17), a general review of 
the work is given here, rather than an outline 
of the work done on fundamental studies only, 
but with particular emphasis on Section 7.1 
activity. The reader will be referred to the 
Summary Technical Report of the other sec- 
tions when appropriate. 

This contract which ultimately led to what 
amounted to the provision of general labora- 
tory facilities for the investigation of a wide 
variety of fire-control problems was initiated 
in June 1941. 

Initially, two kinds of activities were con- 
templated: (a) assistance on specific optical 
and photographic problems which arose in the 
course of other investigations by Section D-2, 
and (b) some long-range general studies of the 
application of light (whether visible or invis- 
ible) to problems of detection, prediction, and 
servo control. Before long a considerable part 
of the contractor's effort was devoted to (c) 
general range finder developments sponsored by 
the group that was later to become Section 7.4 
(see Volume 2). Still later, there was a large 
amount of development on four specific sub- 
jects, in addition to those already outlined; the 
first three were the responsibilities of Section 
7.1 and the fourth of Section 7.3. 

1. Illuminated sight Mark 14. 

2. Stereoscopic observation instrument T7. 

3. Intermediate range director for the 
40-mm gun. 

4. Applications of gyros with pneumatic 
restraint and take-off to various fire-control 
instruments. (See Chapter 4.) 

Optical and Photographic Problems 

Among activities of class (a) were (1) modi- 
fication of the phototheodolites at Fort Monroe 
to give increased precision, (2) design of 
tracking cameras for the study of azimuth and 


elevation errors, (3) design and construction 
of an acuity testing instrument, and (4) con- 
struction of reflecting end windows to permit 
ranging on the sun with optical height finders. 
(See Volume 2.) 

Light as a Working Medium 

In class (b) studies were undertaken by Sec- 
tion D-2 along the following lines : an extended 
base (2- or 3-station) range-finding system;^"’ 
an optical servomechanism and several unit 
mechanisms for use in a director, such as an 
optical memory device, an optical pointer 
matching device, an “optical cam” mechanism, 
and a high-speed electronic multiplier.^^ 

General Range Finder Developments 

In class (c) considerable preliminary work 
was done toward improving the large (13V2- 
ft) precision range finder,^® but early in 1943 
this work was taken over by an Army-Navy- 
NDRC steering committee for the “super range 
finder,” undertaking, in cooperation with the 
Eastman Kodak Co., Keuffel & Esser, and 
Bausch and Lomb Optical Co., a basic redesign 
of such range finders. This activity will be re- 
ported by Section 7.4. (See Volume 2.) 

The majority of the range finder work of 
Project 17 has been in the development of a 
number of short-base range finders with auto- 
collimating features, and mostly of the super- 
imposed coincidence type. The first one, which 
served as a prototype for the others, was a 15- 
in. superimposed coincidence instrument.^®'^® 
This was tried in several modifications for va- 
rious purposes, including tracer crossover ob- 
servation for machine gun sight adjustment 
both in plane-to-plane fire and ground-to-plane 
fire.2^ For various reasons none of these was 
successful. A number of the 15-in. instruments 
were delivered to the Navy, principally for 
trial in connection with the Navy’s gunsight 
Mark 14. 

A second embodiment of the superimposed 
coincidence principle is the 30-in. range finder 
for feeding range into directors of the inter- 
mediate type.2® After several modifications this 
emerged as an instrument of 30-in. base length 




FUNDAMENTAL STUDIES 


23 


and 8 power, with color differentiation between 
the two images. The ranging mechanism is in- 
terconnected with the elevation handwheel of 
the director so as to make it belong to the 
“height finder’' rather than to the “range 
finder” class. It is provided with two eyepieces, 
one of which is used by the ranging operator 
to keep the images matched, and the other by 
an observer who watches tracer crossover and 
introduces the necessary spots into the direc- 
tor. This instrument was incorporated in direc- 
tor M5A1E1 (Project 31) and tested by the 
Antiaircraft Artillery Board at Camp Davis. 
The results were especially promising and, 
hence, six additional directors were equipped, 
delivered, and field tested. The director includ- 
ing this range finder (now standardized as 
MIO) has been standardized as M5A2 and a 
large quantity was built. 



Figure 10. The “fly’s eye,” Mark 14 illuminated 
sight without 45-degree reflector plate. 

Two 48-in. range finders, almost identical 
with the 30-in. director instrument mentioned 
above, were constructed for use with the inter- 
mediate range directors.^® One of these is in- 
stalled on director T28 developed under this 
contract. Other models were designed, particu- 
larly for infantry use, but were not standard- 
ized, although two models (designated T25 and 


T26) were constructed and extensively tested 
before, during, and after several modifications. 
T25, as finally modified, appeared very promis- 
ing and nearly ready for standardization. 



Figure 11. Stereo aid for Maxson turret. 

Illuminated Sight Mark 14 

Under subject 1 was developed a refiecting 
sight known as the “fiy’s eye” (Figure 10) 
because of its multilens nature, the advantage 
of which was that the freedom of the gunner’s 
head movement was greatly increased.^^ This 
will be reported in more detail by Section 7.2 in 
Volume 3. 

Stereoscopic Observation Instrument T3 

Under subject 2, a stereoscopic aiding device 
has been developed for increasing the accuracy 
of tracer fire control with automatic weapons.” 
Considerable work was done on this subject by 
the Polaroid Corporation under Project 32 
(see Section 2.6.1). The first model by the 
Eastman Kodak Co. was unsatisfactory, chiefly 
because of the combined effect of low light 
transmission and excessive vibration. The East- 


24 


LAND-BASED FIRE CONTROL 


man Kodak Co. constructed another model em- 
ploying a new optical system which showed 
promise of being much better under the ex- 
cessive vibration present. This model was tested 
at Fort Bliss and proved to have two diffi- 
culties. The most serious difficulty was that the 
exit pupil was only about 2 mils larger in 
diameter than the usual eye pupil size. Despite 


.51-caliber 4-gun turret Ml at 400 to 600 yards 
(Figure 11). 

Intermediate Range Director for the 
40-mm Gun 

Under subject 3 an intermediate range direc- 
tor T28 (Figure 12 A) was developed.^^ One of 



SPOTTING EYEPIECE 


TRAVERSE SPOT — 

MEMORY TIME SWI 

HEIGHT KNOB 

ELEVATION 
HANDWHEEL 

ELEVATION SPOT 

SLEWING HANDLE 

MEMORY POINT'S! 

MAIN POWER SWITCH 

POWER CABLE PLUG 

GUN CABLE PLUG 


AZIMUTH 

TRACKING SWITCH 


AZIMUTH 

TELESCOPE 


AZIMUTH 

HANDWHEEL 


ORIENTING CLUTCH 
LEVELLING KNOB 

REMOTE RANGE 
PLUG 


Figure 12A. T28 director. 


all the work done to eliminate vibration there 
remained some vibration in the instrument as 
well as some in the observer’s head. The other 
difficulty was due to an error in the optical sys- 
tem which resulted in some curvature in the 
field. Another model was constructed with 
large exit pupil and a flat field. The results of 
firing trials were very satisfactory with range 
up to about 700 yards, beyond which the stereo 
perception rapidly became unsatisfactory. Sev- 
eral small OQ planes were shot dow^i with the 


the Eastman engineers suggested the use of a 
sphere with three mutually rectilinear dipole 
windings^'^’^® as a combined resolver, lead com- 
puter, and synthesizer (Figure 12B). The 
sphere is turned by present coordinates, and 
the leads inserted simply by suitably energizing 
the dipoles. The proper locations for the gun 
azimuth and elevation axes are thus deter- 
mined (Figure 13). A serious difficulty in de- 
termining the instantaneous rates was en- 
countered. At NDRC’s suggestion, therefore, 




AUTOMATIC WEAPONS 


25 


the attempt to obtain instantaneous rates was 
abandoned, and the project proceeded on the 
memory point system. 

The preliminary tests were completed at 



Figure 12B. “Magic Ball” or combined resolver, 
lead computer, and synthesizer making use of 
three mutually rectilinear dipole windings, 
wound on sphere. 

Fort Bliss, Texas, and disclosed a number of 
troubles and errors that needed to be corrected, 
but the termination of World War II stopped 
further development. 

Pneumatic Controls 

Under subject 4, Section 7.3 carried out a 
series of considerable developments of pneu- 
matic computing systems, which will be re- 
ported in more detail by that section. (See 
Chapter 5.) 

2 ^ AUTOMATIC WEAPONS 

Development of Intermediate 
Director M5A2 

The third category of Section 7.1 activity 
listed above (see Section 2.1.2) was the devel- 


opment of automatic weapons. This work was 
in reaction to the very unsatisfactory state of 
40-mm fire control. It cannot be said that a 
complete and satisfactory system had been 
developed even at the termination of World 
War II. At the beginning of the war the 
standard director for control of 40-mm fire was 
the M5 (Kerrison) director which depended 
upon certain observations of the tracers, the 
condition for such observations being very dif- 
ficult to obtain in the field, particularly against 
high-speed targets. Consideration was given 
early in 1941 to this problem and several steps 
were taken at various periods to improve this 
situation. 

Barber-Colman Co. (Project 31) 

Contract OEMsr-268 was instituted at the 
Barber-Colman Co. which sought to provide a 
better working, simpler, and more easily pro- 
curable modification of the M5 director. The 
principal substitutions were eddy current drag 
disk governed motors for the ball integrators 
and torque amplifiers, and an electronic multi- 
plier for the mechanical multipliers. The direc- 
tor, T21, was completed, but testing at Camp 
Davis showed no improvement in hits, and the 
saving in cost was judged insufficient to war- 
rant changing. Furthermore, the most trouble- 
some feature of M5 had meanwhile been cured 
by the elimination of the torque amplifiers and 
substitution of larger and higher speed ball in- 
tegrators as the variable speed drives. 

In order to try any improved solution, it was 
necessary to have a servo for the 40-mm gun 
which would follow with reasonable accuracy. 
The existing servo had a very bad velocity lag 
— as much as 2 degrees at high speeds. The 
contract contained authority to build a better 
servo, and a clutch-type unit was constructed 
and used for many of the tests. 

As the next step the Eastman 30-in. red- 
green range finder developed under Project 17 
was mounted on the director. This range finder 
provides the tracer crossover method of spot- 
ting. This combination did not give satisfactory 
results, principally because it was difficult to 
train a ranging operator to follow the rapidly 
changing range. Aided tracking was provided, 


SONFIDENTIA: 


26 


LAND-BASED FIRE CONTROL 


but not only did the range change rapidly but modified M5 was built mounting the Eastman 

also the rate at which it changed. Also, the range finder and having means by which range 

operator could not satisfactorily range and would be automatically driven into M5. With 

adjust the fire at the same time. this modification spotting in range was easily 



prebIcti 


60 CYQLf MEMORY^ 1 






c 


EtEVATIO 
RACKING U 




AZIS'ht TRACKING UNIT, 
AZIMUTH DRIVE GEAR 


ELEVATION SECTOR 


Figure 13. Interior view of T28 director. 


Two changes were then made, one to permit 
the operator to match range by setting height 
instead of slant range and the other to provide 
two eyepieces so that a second operator could 
adjust the fire. Angular height was obtained 
from the director and slant range automatically 
computed and power-driven into the director. 

Inasmuch as it was desirable to convert M5's 
rather than build completely new directors, a 


provided and the method of operation was very 
similar to that employed with M5. The differ- 
ences (and they were verj’ important ones) 
were: nearly correct range continuously sup- 
plied to the new director designated as 
M5A1E1,2‘ and an opportunity to see on every 
shot the direction and amount of necessary cor- 
rection, whereas with M5 one can only tell the 
direction of correction, and then only on rare 






AUTOMATIC WEAPONS 


27 


occasions when a line-of-sight shot is obtained. 

Six M5A1E1 directors were delivered to the 
Army on June 20, 1943. These six units were 
sent to the South Pacific, and very favorable 


kno\Ning the exact direction in which to push 
the lever. 

Another modification having separate azi- 
muth and elevation spotting arrangements was 


TRACER OBSERVERS RANGE RCS HEIGHT RANGE FINDER OPERATORS 

EYEPECE FINDER ADJUSTMENT ADJUSTMENT EYEPIECE 


RANGE SCALE 
LIGHT 



RANGE 

ADJUSTMENT 


RANGE 

HANDWHEEL 

(removable) 


RANGE FINDER 
SWITCH 


DRECTOR RANGE 
SWITCH 


: 1 

NGE SCALE LIGHT 
INTENSITY ADJUSTMENT 


Figure 14. M5A2 director. 


informal reports of their performance were re- 
ceived. The director was standardized as 
M5A2, and the production order was given to 
Singer Manufacturing Co. (Figure 14). 

The Antiaircraft Artillery Board requested 
that a model be supplied them for test embody- 
ing modifications to see whether spotting in 
elevation and azimuth is feasible by means of 
a joystick hand lever. With this arrangement 
both trackers continuously track on the target. 
The model including the joystick hand lever 
was tested but proved to be little or no im- 
provement, for the spotter had difficulty in 


also tested and appeared to be as good as, or 
possibly a little better than, the standard range 
spotting used on M5A2. It is doubtful that the 
improvement is great enough so that the Anti- 
aircraft Artillery Board \rfll recommend a 
change. 

The Antiaircraft Artillerj" Board also re- 
quested that a standard director M5A2 be 
modified so that it would accept radar inputs. 
The director was delivered to Fort Bliss in 
June 1945, but when supplied with data from 
SCR-584, it was found that the perturbations 
of SCR-584 data at about one cycle per second 


CONFIDENTIAL 


28 


LAND-BASED FIRE CONTROL 


were considerable and that a smoother had to 
be introduced between SCR-584 and M5A2.28 

A series of mathematical studies^® on gyro- 
scope substitutes were carried on under this 
contract for comparison with standard gyro- 
scopes; one was a fluid gyro and the other a 
vibrating reed tachometer. Both proved of in- 
terest but of insufficient advance over standard 
gyros to warrant an intensive program. (See 
Chapter 3.) 

Two other projects contributed to the devel- 
opment of M5A2: the Project 17 (Eastman 
Kodak Co.) contribution of a suitable range 
finder cited above, and an experimental investi- 
gation carried on at the GM Laboratories, Chi- 
cago, Illinois. 


2.4.3 Qjyi Laboratories, Inc. (Project 26) 

Under contract OEMsr-184 the GM Labo- 
ratories was requested to develop an electrical 
lead-computing device for intermediate range 
directors, and a servomechanism for convert- 
ing the lead angles from the electrical to the 
mechanical form was accordingly undertaken. 
By the use of suitable tapered potentiometers, 
and with a stable electronic amplifier and in- 
duction motor combination, the desired ac- 
curacy of plus or minus 0.5 per cent between 
the input and output rotary motions was ob- 
tained. A model incorporating both elevation 
and azimuth systems was built and with some 
modifications was used in the standard direc- 
tor M5A2.29 (ggg Chapter 3.) 


^ Development of Intermediate 

Director M7 (Weissight) 

A schematic of a course-and-speed emer- 
gency sighting system adaptable for use with 
the 40-mm antiaircraft gun or the Maxson 
turret was drawn up,** and a diagrammatic 
model was made to show the method of opera- 
tion. The system is a mechanical vector solution 
in which course and speed are set by estima- 
tion, but it is stabilized in azimuth so that once 
the course is correctly set it will remain correct 
so long as the plane flies in a straight line, 
b By H. K. Weiss of the Antiaircraft Artillery Board. 


Development at the Pitney-Bowes 
Postage Meter Co. (Project 61) 

A preliminary model for the 40-mm Bofors 
was built under contract OEMsr-883 at the 
Pitney-Bowes Postage Meter Co. and tested at 
Camp Davis. As a result of comparative tests 
between this sight, a British version of course 
and speed sight known as the Stiffkey stick, 
the forward area sight, the Weissight was 
standardized as M7 for use on the 40-mm gun, 
built in considerable quantities, and used 
throughout the remainder of World War II. 

With this device an additional operator is 
employed whose sole duty is to set course and 
speed. The usual two trackers are thus enabled 
to concentrate on following the target without 
diverting their attention to estimating lead 
(Figure 15).®® 



Figure 15. Course and speed sight, M7. 


The course and speed sijiht is set by pivoting the arrow 
(located on top of the box in tbe extreme right of the pic- 
ture) along target path, and setting the dial on the box 
in accordance with estimated target speed. The telescopes 
are then used to track the target. The gun will “lead” the 
line of sight in accordance with the problem by virtue of 
geometrical links between the gun and the telescopes. The 
forward area sights (cart wheels) are stand-by, or emer- 
gency, sights and should not be confused as part of 
Weissight system. 


Course-Invariant Sights 

Perhaps one of the most serious problems in 
connection with the use of gyro lead-computing 
sights (see Chapter 4) is that an accurate and 
continuous range is necessary in order to have 


'^<; ^NFIDENTI^ 


AUTOMATIC WEAPONS 


29 


a good solution, and such a range is not usually 
obtainable. Inasmuch as the rate of change of 
range often is rapid and its derivatives also are 
rapidly changing, the possibility of estimating 
range correctly and continuously is very slight. 
At present the most satisfactory arrangement 
appears to be to set a range through which the 
target will fly. This system is moderately satis- 
factory against directly approaching targets 
but is not satisfactory against crossing targets. 
In view of this difficulty, a type of lead-comput- 
ing sight without range input was proposed,'" 
by which invariants of the course can be in- 
serted and, by appropriate mechanisms, con- 
tinuous range produced for and introduced into 
the computation. Thus, in using such a sight, 
the operator might attempt to refine his esti- 
mate of invariant during tracking either by 
observing tracers or by comparing the range 
indicated in the computing mechanisms with 
the true range at some point of the course. The 
advantages to be gained by such a sight led 
Section 7.1 to negotiate a contract for its devel- 
opment. 


* ^ ^ Development at Baker Manu- 
facturing Co. (Project 73) 

Accordingly contract OEMsr-1190 was nego- 
tiated with the Baker Manufacturing Co. for 
studying methods of adapting gyroscopic sights 
to the invariant system of lead computation. A 
proposal for an invariant gyroscopic sight was 
made by the company and was discussed in a 
conference at the Applied Mathematics Panel.^^ 

This development can be considered as a new 
approach to automatic weapons fire control 
which will permit one-man operation (Figure 
16) and which, in very general terms, combines 
the desirable qualities of the gyroscopic lead- 
computing sight and those of the vector sight.^^ 
The new sight is on the carriage, and in instal- 
lation and in tracking of the target more nearly 
resembles a gyro lead-computing sight than a 
separate director such as M5. In more detail, 
an angular lead is produced from a processed 

c Independently by H. H. Germond of the Applied 
Mathematics Panel and H. K. Weiss of the Antiair- 
craft Artillery Board. 


gyro as in the gyro lead-computing sight, but 
the lead angle, instead of being proportional 
to angular velocity and to set-in range, will be 
such a function of the angular velocity that a 





Figure 16. Invariant gyroscopic lead-computing 
sight mounted on M45 turret (gun removed). 


one-knob adjustment once correctly made for a 
given unaccelerated target course remains cor- 
rect for the remainder of this course. 

The optical arrangement was decided upon 
in conferences with the Eastman Kodak Co., 
and this section of the mechanism actually was 
built by Eastman under Project 17 referred 
to above. 

The sight, completed after the termination 
of World War II, was mounted on a Maxson 
turret and some testing completed. Vibration 
troubles had been expected and an attempt 
made to guard against them, but when four 
guns of the Maxson turret were fired the mount 
vibrated so badly that firing tests could not be 
conducted. Considerable development appears 
to be necessary to overcome this difficulty. The 
principle itself appears to be promising, and it 
is thought that the project should be continued 
by the Army. 


ijnfidential: 


30 


LAND-BASED FIRE CONTROL 


* Intermediate Range Director T28 

Under the broad fire-control research pro- 
gram at the Eastman Kodak Co. (Project 17) 
(see Section 2.3.7), there was developed the 


OD-56) for the development of a director for 
controlling a battery of 3-in. antiaircraft 
rocket projectors. The director was to be use- 
ful against high- and low-level targets, as well 
as against diving targets, and provision was to 



Figure 17A. A A rocket director T18 side view, covers removed. 


novel intermediate range director T28 for the 
40-mm gun. The most particular point of nov- 
elty resides in the use of an electromagnetic 
dipole vector solver. The reader is again 
invited to review the Eastman reporU^ for 
details of this important development. 

25 MISCELLANEOUS WEAPONS 

* * ^ Antiaircraft Rocket Director 

(Project 38) 

Contract OEMsr-517 with the Bristol Com- 
pany was negotiated in response to a request 
of the Ordnance Department (Directive 


be made for operation with either optical or 
radar inputs. 

The apparatus consisted of two separate 
parts — a tracker and a computer (Figure 
17 A). The tracker (M5 modified) was to be 
used for nearby targets, whereas the tracker 
and the computer was to be used for distant 
targets. The tracker included the usual optics 
and their aided drives, together with transmit- 
ting selsyns to get the data to the computer. 
In addition, the direct angular prediction 
mechanism included in the tracker was for use 
against near and diving targets, and the same 
seises transmitted to the projectors directly. 


MISCELLANEOUS WEAPONS 


31 



Figure 17B. AA rocket director T18, top view showing charts: waxed (lower), and ballistic 
(upper). 


The computer incorporated three recording 
voltmeters, and rates were obtained by match- 
ing the slopes of the lines. Inasmuch as con- 
tinuous fire was not contemplated no feedback 
of time of flight was necessary, and the solu- 
tion was extremely simple. The ballistics were 
included, only partly mechanized, i.e., a chart 


reading (Figure 17B) had to be set on a dial. 
Selsyns transmitted the information to the 
projectors.'^^ 

The director was completed and trials at 
Camp Davis were finished, except for firing. 
No ammunition was available, and the projec- 
tors were not sent to Camp Davis. Operation 




32 


LAND-BASED FIRE CONTROL 


of the director was satisfactory and the instru- 
ment itself was in form suitable for production. 

At this point NDRC was informed that there 
was no longer a requirement for rocket projec- 
tors and, therefore, none for the director. 

A modification of this director for use as a 
data smoother and retransmitter between 
SCR-268 and M7 was built under Project 64. 

* Antitank Computing Sight T62 
(Project 59) 

At the request of the Ordnance Department 
a lead computer for the 75-mm gun mounted 



Figure 18. Antitank lead-computing sight T62. 


on the motorized gun carriage M3 was devel- 
oped at the Barber-Colman Co. under contract 
OEMsr-892.34 

The computer (Figure 18) offsets the tele- 
scope from the gun bore axis in accordance 
with (1) lead angle in azimuth, (2) azimuth 
correction due to tilt of the gun trunnions, and 
(3) super-elevation correction. The azimuth 
lead angle is computed by the memory point 
method, the azimuth travel of the target being 
measured for a time equal to the (estimated or 
measured) time of flight. An advantageous 
feature of this design is that as the lead angle 
changes due to change of range and angular 


rate, the telescope displacement is altered only 
by the change in lead angle. Another action of 
this character restores the telescope to align- 
ment with the bore before a new computation is 
made. 

Successful firing trials of the computing 
sight T62, mounted on the gun carriage M3, 
were conducted at Aberdeen, April 25 through 
May 1, 1944, and at the Tank Destroyer Board, 
Camp Hood, Texas, August 12 to 14, 1944. The 
weapon was turned over to the Tank Destroyer 
Board as a museum piece at the request of the 
Ordnance Department. 

2^ SHORT-BASE RANGE FINDERS 

Reference has already been made to the 
range finder developments at the Eastman 
Kodak Co. sponsored by the division, which 
was one phase of a very broad contract set up 
with that company for the development of fire- 
control instruments. In the interests of inte- 
gration, all activity of Section 7.1 at the East- 
man Kodak Co. was reported in Section 2.3.7 
with a synopsis given of the work sponsored 
by other sections. In addition to this work done 
under the section’s auspices on short-base 
range finders two further projects are to be 
reported. 


2 . 6.1 Polaroid Corporation (Project 32) 

Contract OEMsr-302 was originally estab- 
lished for the purpose of developing short- 
base range finders suitable for use in con- 
nection with intermediate range antiaircraft 
guns and plane-to-plane fire control. * Under 
Bureau of Ordnance Directive NO-112, a 43-in. 
stereoscopic range finder of very simple design 
was worked out and 25 were constructed for 
trial. A comparative test of these and the 15-in. 
range finder, developed by Eastman Kodak Co. 
under Project 17, showed the two instruments 
to be of about equal precision when used by 
unselected and untrained personnel. In the 
hands of carefully selected and trained person- 
nel the Polaroid range finder was more precise. 

The Bureau of Ordnance later requested that 



TESTING PROGRAM 


33 


twelve 43-in. range finders be equipped with 
mechanism for setting range continuously into 
the Mark 14 lead-computing sight. The first of 
these devices was in combat service and the 
second was later completed and used for dem- 
onstration at Dam Neck on February 23, 1944. 
The design and construction of an improved 
model employing the motor servo was started, 
but this was discontinued because the results 
of the first unit in combat service indicated 
that there was not sufficient time to make the 
necessary observation, and it was difficult to 
keep the sight operator and the range finder 
operator on the same target. It was concluded 
that the benefits which could be derived from 
such a device were not sufficient to justify 
further work on the project. 

The stereo aid project, discussed under Proj- 
ect 17, was initiated at the Polaroid Corpora- 
tion and the first model constructed by them. 

Several other ideas for short-base range 
finders were investigated, but the development 
of them was not carried very far. They are 
related in the Polaroid final report.^® 

Combined Tracking and Range- 
Finding Devices (Project 52) 

Contract OEMsr-735 was set up with the 
Barber-Colman Co., contemplating the design 
and construction of two combined antiaircraft 
tracking and range-finding devices. One was to 
be for a 30-inch-base range finder and one for 
a 13 1 / 2 -foot-base range finder. Some work was 
done on each but neither was constructed; in 
fact, this project was changed a number of 
times. 

The range finders which were to be used as 
an integral part of these tracking devices were 
those (self-collimating) which were developed 
under NDRC auspices at the Eastman Kodak 
Co. and reported above under Project 17. 

At one time, the unit which was to have a 
131 / 2 -foot range finder was to be redesigned be- 
cause of an urgent need for a tracker for the 
director T15, the principal object being to 
provide a substantially lighter and cheaper 
tracker than the one used with M9 and com- 
bined with a range finder.^s The potentiometers 
and associated mechanisms required in the M9 


tracker are obviously not required for T15. 
The layout was substantially completed, and 
then all work was stopped because T15 was not 
standardized. 

2 7 TESTING PROGRAM 

At the time the NDRC started its work, one 
of the apparent lacks in the equipment of the 
Army and Navy was an adequate program for 
testing antiaircraft fire-control systems. Fur- 
thermore, there was in existence very little test 
equipment to carry out such a program. 

^ ^ ^ Heavy Antiaircraft Fire Control 

Probably the most straightforward problem 
was that of testing heavy antiaircraft fire con- 
trol. This was true because the devices which 
constituted the system were largely automatic, 
thus making unnecessary tests involving 
human elements. For test purposes the fire- 
control system was divided into three parts, 
namely the tracking equipment, the predictor 
or computer, and the gun. The computer was 
selected as an appropriate subject for investi- 
gation because of its importance in the chain 
•and because new computers were being pro- 
posed and developed. 

2 72 Dynamic Tester (Project 25) 

The first device designed particularly for 
testing computers dynamically (Figure 19), 
that is, in motion as if they were actually 
operating, was developed under contract 
OEMsr-98 with the Barber-Colman Co. This 
machine was outlined and proposed by the 
Barber-Colman Co. in order that information 
could be obtained rapidly about two things: 
first, the errors committed by a computer when 
the data supplied to it was essentially perfect; 
and second, the manner in which the computer 
would handle certain types of perturbation in- 
troduced by the imperfections of the tracker.^^ 

The perfect data for the Barber-Colman 
dynamic tester was stored by cutting a set of 
very accurate cams. Servo motors associated 
with three of the cams were provided to drive 
the handwheels of the computer under test 


]C 0?^1DE.M~IAL y 


34 


LAND-BASED FIRE CONTROL 


exactly as if a perfect tracker were following 
an idealized target. The correct gun orders for 
this target were stored on the remaining three 
cams and compared electrically with the gun 


curved flight on the part of enemy targets was 
encountered, it became clear that a great 
variety of test courses would be necessary. 

The dynamic tester developed under Project 


PERTURBATION FREQUENCY 
ADJUSTMENT 



FLYWHEEL 


CLUTCHES* 


OUTPUT SHAFTS 


CAM 


CLUTCH DRIVE MOTOR 

/ 


CLUTCH CONTACTS 

FRICTION ADJUSTMENT / ^CAM SHAFT GEAR 


STOP PIN 


SYNCHRONOUS MOTOR 


STARTING POINT 
ADJUSTING PIN 


ECCENTRIC 

ADJUSTMENT 


Figure 19. Barber-Colman dynamic tester range unit. 


orders actually transmitted by the computer. 
Errors in the gun orders were recorded elec- 
tically on graph paper. 

The dynamic tester was found to be of great 
value, since many runs could be made in a 
short time and the errors observed instantane- 
ously. A perturbation unit added sinusoidal 
variations about the perfect tracking data and 
was so arranged that the effects of various am- 
plitudes and frequencies could be observed. It 
is readily apparent how important such a rapid 
evaluation of errors can be in the expeditious 
development or modification of complicated 
systems. 

2.7.3 Punched Tape Dynamic Tester 
(Project 60) 

When in the later stages of World War II 
the problem of coping with evasive action and 


25 was limited as to the number of courses 
available by the fact that each new course re- 
quired a set of input and output cams, which 
were rather expensive. Also, the process of 
changing from one course to another required 
several hours. 

To simplify the process of preparing data 
for use by a dynamic tester a new mechanism 
was outlined and proposed by Section 7.1, and 
contract OEMsr-904 with the Bell Telephone 
Laboratories was undertaken to develop a 
dynamic tester in which the data for courses 
is supplied by punched tapes which could be 
prepared and interchanged with a great saving 
in time and expense.^® 

A breadboard model of one unit of the tape 
tester was first constructed to prove the prac- 
ticability of the idea. A complete tester, called 
the Model I tester, with three input and three 
output channels (Figure 20) was then built 


TESTING PROGRAM 


35 


and used at the Bell Telephone Laboratories 
for the tests on curvature modification of di- 
rectors T15 (Project 30) and M9 (Project 
78), discussed above. Later, the Model I tester 



Figure 20. The punched tape dynamic tester — 
complete assembly. 


was used for a number of other projects and 
was finally delivered to the Antiaircraft Artil- 
lery Board. 

The testing of naval directors such as the 
new Mark 56 calls for a dynamic tester having 
a larger number of inputs and higher accelera- 
tions resulting from the ship’s motion (see 
Chapter 6). A model was accordingly devel- 
oped having six inputs and capable of accelera- 
tions of the order of 1,000 mils per sec^ with 
a probable error of about 1 mil. It has three 
output channels utilizing a “pseudosynchro” 
capable of making spot checks of the outputs 
at 0.3-sec intervals, the errors being recorded 
on recording meters as in the Model I instru- 
ment. 

A bench model was set up during the sum- 
mer of 1944, and the first completed instru- 
ment was expected by January 1945. An un- 
avoidable delay was brought about by a pro- 
longed search for the cause of faulty operation 
of the bench model. The first Model II instru- 
ment was to be delivered to the Radiation 
Laboratory for the Mark 56 tests, but on ac- 
count of changes in plans occasioned by the 
end of the war, this model was delivered to the 
Naval Research Laboratory. A second Model 


II was under way for the British, but the dis- 
continuance of lend-lease made it necessary 
to cancel the order for this instrument. 

An auxiliary device called the Relay Inter- 
polator was built at the suggestion of Section 
7.1 under Section 7.5 auspices (Project 70) to 
facilitate the preparation of the tapes at the 
Bell Telephone Laboratories, under contract 
OEMsr-1160. A summary report by Section 
7.5 on this device will be found in Chapter 5. 

Although the dynamic testers could simulate 
tracking data from any of the standard sources, 
the Army prefers to have a part of the test 
run with the computer receiving actual track- 
ing data at the testing ground. When such 
tests are run, difficulties of recording data and 
of calculating the errors are such as to over- 
load the computing facilities available for the 
work. Unlike the dynamic tester tests, this 
method cannot reproduce courses exactly. 
Hence much more data must be obtained in 
order to derive reliable results. 


Data Recorder and Ballistic 
Computer (Project 63) 

Mechanisms were built under Division 7 
contracts to ease the situation. The first of 
these was a data recorder proposed by Section 
7.1 and built by the Bell Telephone Labora- 
tories under contract OEMsr-965.^® It ac- 
cepts standard selsyn data from the tracker 
and the computer, translates and prints these 
quantities in numerical form on six tapes. The 
data are thus immediately available. Normally 
a record is made at intervals of 1 sec within 
0.004 sec or less and records to the nearest 1/2 
mil or to the nearest yard. The mechanism 
provides for each channel a selsyn-servo com- 
bination, a gear train with commutators which 
set up digital codes on relays, and a standard 
“ticketer” or printer, together with such con- 
trol relays as may be required. 

To shorten the computing time a ballistic 
computer was built at the suggestion of Sec- 
tion 7.1 under Section 7.5 auspices at the Bell 
Telephone Laboratories under contract OEMsr- 
1236. This device to be reported upon by Sec- 
tion 7.5 (see Chapter 5) carries through auto- 





36 


LAND-BASED FIRE CONTROL 


matically all the operations needed to calculate 
the correct times of flight and gun orders for 
the data observed in the tests, and prints the 
errors committed by the predictor. It does the 
work of about 50 girls with standard calculat- 
ing machines. 

Liaison with the Antiaircraft 
Artillery Board (Project 54) 

Testing of the various antiaircraft devices 
developed by the Fire-Control Division of 
NDRC was carried on chiefly by the Antiair- 
craft Artillery Board. A very close working 
relationship was maintained between NDRC 
and the Board, and contract OEMsr-767 with 
the University of North Carolina provided the 
services of an NDRC engineer,'^ who was sta- 
tioned permanently at the Antiaircraft Artil- 
lery Board location. 

In addition to assisting with the testing of 
all NDRC equipment at the Antiaircraft Ar- 
tillery Board, and assisting the Board itself 
whenever asked to do so, the work involved 
the design and construction of a considerable 
number of instruments and pieces of test equip- 
ment, many of which have been built in the 
University of North Carolina shops. In this 
way it was possible to obtain quickly such 
items as would otherwise hold up the testing 
of important antiaircraft equipment. At times 
extra personnel was obtained from the Univer- 
sity of North Carolina. 

Apparatus constructed at the University of 
North Carolina included the following major 
items : 

1. A film-type slide rule suggested by Sec- 
tion 7.1 for use in speeding up the necessary 
trigonometric computations involved in direc- 
tor testing.^® 

2. Apparatus for use with the precision kine- 
theodolites. 

3. Stereo photographic apparatus for assess- 
ment of Are in which tracer is used. 

Automatic Weapons Assessment 

The development of dynamic testing equip- 
ment for directors gave NDRC a satisfactory 

^ Paul Mooney. 


means of comparing fire-control apparatus for 
guns as distinguished from automatic weapons. 
These dynamic director testers could be, and 
were, modified to make dynamic tests of cer- 
tain types of automatic weapons fire-control 
equipment, but they were not suitable for fire- 
control apparatus which involves the adjust- 
ment of fire by observation of the tracer. Con- 
siderable thought was given to the possibility 
of constructing a device similar to that made 
by Section 7.2 at the University of Texas for 
airborne gunnery assessment (see Volume 3). 
In this device an optical target is presented to 
the gunner by means of an optical system 
mounted on a car which moves up and down 
a crescent shape track. The track itself can 
be moved about a vertical axis. The automatic 
weapons problem is complicated, first, by the 
fact that equipment being tested is several 
times as large as that used for airborne gun- 
nery, and, in addition, it should be appreciably 
more accurate ; furthermore, the fire control of 
automatic weapons depends upon tracers and 
it would be necessary to simulate at least part 
of the tracer path in order to obtain a true 
assessment. A mechanism was proposed which 
appeared to meet all the requirements and to 
be at least worthy of further study. 

The Armour Research Foundation was con- 
sulted and undertook a preliminary study to 
determine : 

1. Whether or not in their opinion the pro- 
posed scheme actually did meet the require- 
ments. 

2. Whether the above scheme had elements 
which were impossible or impracticable me- 
chanically. 

3. The time and effort required to construct 
the mechanism. 

4. Whether any alternatives which might be 
better or cheaper or more practical could be 
found. 

The answer to item 1 turned out to be that 
the scheme would meet the requirements, and 
to item 2 that the elements were possible and 
probably practicable but some of them quite 
difficult to accomplish. 

However, the project turned out to be so 
large (several hundred thousand dollars and 
three years’ time would be required) that it 


TESTING PROGRAM 


37 


seemed unwise to undertake it at the time, and 
the Armour group was unable to find any modi- 
fication which would meet the requirements 
and yet would take substantially less time and 
effort. 

The problem of testing AW fire control is 
one, however, which should be given consid- 
erable thought in the future. The present 


method of firing is of course unsatisfactory 
because of the difficulty of coordinating the 
flight of a test target with the activities on the 
ground, the dependence of such tests on 
weather conditions, the expenditure of ammu- 
nition and fuel and inability of a target towed 
by a plane to reproduce courses or to simulate 
high-speed or evasive flight. 


ONFIDENTIAL 


Chapter 3 

SERVOMECHANISMS 


ORIENTATION 

T he importance of servomechanisms in fire- 
control devices was recognized at the found- 
ing of the Fire-Control Subcommittee (Sec- 
tion D-2, later to become Division 7) of the 
National Defense Research Committee. In 
fact, the preliminary agenda for the initial 
meeting of the committee (dated August 1, 
1940) stressed as one line of activity for the 
group, basic program of development of 
servomechanisms.” Hence it is not strange to 
find that the first and second of the Reports 
to the Services issued by the Fire-Control Sub- 
committee were devoted to this subject,^'^ re- 
sulting from the initial contract of Section D-2. 

At the time of writing of this Summary 
Technical Report there has already appeared 
a number of treatises on servomechanisms 
which represent the state of the art at this 
date (1946). Several documents have been is- 
sued by Division others have been pub- 

lished by commercial houses. 

One of the latter is an excellent and concise 
treatise® written at the request of Division 7 
and the Applied Mathematics Panel of NDRC. 
Because of the timeliness of this book and its 
position relative to the NDRC activities, little 
need be recounted here as to the scope, impor- 
tance, and present state of the servomechanism 
art. Indeed, a servomechanism will not even 
be defined here, since the Foreword and the 
first three chapters of the book® are devoted 
to this very task. 

Certainly these few words summarizing the 
state of the servo art would be incomplete 
without a reference to the long history of 
automatic controls leading to the development 
of the modern servomechanism. A recent pa- 
per® provides an excellent bibliography on 
controls and leads one to the present from the 
centrifugal governor invented by Huygens as 
a possible means of regulating a clock and 
which was subsequently adapted by Watt to the 
speed control of the steam engine. In this paper 
are reviewed the various theories of stabiliza- 


tion of control devices starting with the rather 
tedious classical treatment to the recent simpli- 
fied graphical methods of the communication 
engineer. 

The following sections summarizing the 
servo activity of Section 7.3 will therefore 
merely restate briefly specific section projects 
within this field. 

3 2 GENERAL ASPECTS OF 

SERVOMECHANISMS 

Of the sixteen contracts which were the 
responsibility of Section 7.3, six were con- 
cerned directly and five indirectly with servo- 
mechanisms. Only one project, the first con- 
tract negotiated by Section D-2, dealt with 
general aspects of the problem; the remainder 
dealt with specific Service problems or fields 
which were felt to be somewhat neglected. 

In 1940 the position of the servomechanism 
with respect to the system of which it is a com- 
ponent was not universally appreciated. This 
was particularly so with regard to military 
applications where secrecy considerations had 
resulted in piecemeal engineering of the sepa- 
rate components of a complete system. Rec- 
ognizing these facts, the Fire-Control Sub- 
committee sought to undertake a substantial 
basic research program. Contract NDCrc-163 
(Project 1) was accordingly initiated with the 
Massachusetts Institute of Technology and five 
problems of a basic nature were outlined for 
solution.^ 

During the course of this program, however, 
a part of the contractor's personnel was 
urgently needed for immediate Service prob- 
lems under other agencies and this more am- 
bitious program was curtailed.^ 

Two reports were issued, however, under 
this contract, although one, a paper on servo- 
mechanisms behavior and design,* was merely 
printed and distributed under NDRC auspices. 
The paper was initially prepared in response 
to an invitation from the American Society of 
Mechanical Engineers, but because of its time- 




38 


SERVOS FOR MEDIUM CALIBER GUNS 


39 


liness and pertinence to the National Defense 
Program it was thought that the material 
should be distributed as a classified publication. 
The other report pertaining to a relay con- 
troller® reported the development of a partic- 
ular device required at that time by the Army, 
namely, an automatic contact-type controller 
which provided a means of replacing the 
manual matching operation necessary for the 
proper operation of the fuze setter M8. 

33 HYDRAULIC BOOSTER SYSTEMS 
FOR SMALL GUNS 

At the request of the Armament Laboratory 
of the Air Corps (Service Directive AC-28), 
Project 15 was undertaken under contract 
OEMsr-18 (later OEMsr-173) was the United 
Shoe Machinery Corp. to develop power 
boosters to aid gunners in maneuvering ma- 
chine guns against the aerodynamic forces on 
the guns. In particular, boosters were desired 
for the flexible single .50-caliber machine guns 
mounted in the rear center of the A20B light 
bomber, and for the twin .50-caliber machine 
gun mount for the tail of the B17E heavy 
bomber. 

The mount developed, delivered for installa- 
tion in the A20B plane, used a velocity-type 
control. Two gun mounts complete with servos, 
delivered for tests in B17E bombers, presented 
two solutions to that problem.® One of these 
mounts had a velocity-type control, the other a 
displacement servo wherein a certain angular 
movement of handgrips resulted in the propor- 
tional angular movement of the gun. Also, this 
latter mount had two azimuth axes, one for 
each of the two .50-caliber guns, in order to 
eliminate torques due to recoil. The other 
mount had a single azimuth axis, the servo 
having to withstand the firing torques. 

The work under contract OEMsr-18 was 
continued under OEMsr-173, and sought to im- 
prove the first A20B hydraulic gun control unit 
developed and to improve also the B17E gun 
control units, including the addition of a panto- 
graph mount for an optical sight and aided 
tracking control.^® With the completion of these 
units and their transfer to the Army Air 
Corps, further work at the United Shoe Ma- 
chinery Corp. on hydraulic boosters for ma- 


chine guns mounted in aircraft was taken over 
by direct contract with the Army Air Corps in 
accordance with an agreement between Section 
D-2 and the Armament Laboratory, Wright 
Field. 

While the project under contracts OEMsr-18 
and 173 called for specific design of boosters 
and hydraulic servos for specific gun mounts, 
a project was initiated with the same con- 
tractor under OEMsr-19 (Project 16) for a 
broader research and development program of 
servos for aircraft gun mounts. It was accord- 
ingly recommended that thought be given to the 
design and construction of means for attaining 
aided tracking with hydraulic servos. Con- 
sideration was also given to the development 
of improved types of variable displacement 
pumps and also to develop hydraulic servos 
suitable for operation from gyroscopes to afford 
lead computation for the firing of guns. Only 
the aided tracking portion of the development 
successfully matured. Further contracts for 
producing a more compact design of booster 
for application to particular Air Corps gun 
mounts were placed with the United Shoe Ma- 
chinery Corporation directly by the Air Corps. 

3^ SERVOS FOR MEDIUM CALIBER 
GUNS 

In May 1942, contract OEMsr-686 was nego- 
tiated with the Westinghouse Electric and 
Manufacturing Co. (Project 46) which repre- 
sented the culmination of development work 
under projects with two other contractors, 
OEMsr-964 (Project 27) with the Barber-Col- 
man Co., and OEMsr-522 (Project 35) with 
the Massachusetts Institute of Technology. The 
Westinghouse contract called for the produc- 
tion design of servos for medium caliber guns.^^ 

® ^ ^ Research at the Barher-Colman 
Company 

For some time prior to the setting up of a 
formal contract with the Barber-Colman Co., 
that organization had carried on at its own 
expense, but with Division 7 engineering assis- 
tance, the development of clutch-type servos.^® 
In April 1943, a contract was negotiated for the 
furtherance of this effort under the heading 


40 


SERVOMECHANISMS 


“Development of Stabilizing Means for Servos.’' 
Although under a broad program, the project 
was rendered concrete by designing the stabi- 
lizing means for a clutch-type servo ; the stabi- 
lizer, however, was of such a nature that it 
could be used with any other servo which has 
an output torque approximately proportional 

t 



Figure 1. Oil Gear M3B1, right side view, show- 
ing electric drive motor (in top recess), bore- 
sighting clutch in engaged position and output 
coupling (lower left). 

to an input of voltage or current. This stabiliz- 
ing means was also an example of “output 
control,” i.e., the stabilizing signal was made 
up of one or more derivatives of the position of 
the output shaft of the servo, in contrast to 
stabilizing means which are derivatives of the 
error or of the input. 

Research at the Massachusetts 
Institute of Technology 

Concurrently with this program a project 
was started at the Massachusetts Institute of 
Technology under Ordnance Directive OD-53. 
The project had two objectives. The first ob- 


jective was to study the Army remote control 
systems Ml for the 37-mm gun mount and the 
remote control system M5 for the 40-mm gun 
mount, and, if possible, increase the output 
torque at low output speeds and eliminate the 
large velocity lag of the system. The second 
objective was to design a new hydraulic remote 



Figure 2. Oil Gear M3B1, left side view with 
covers removed and transparent windows substi- 
tuted to show interior. Slewing clutch lock and 
lever and output coupling are at lower right. 
This gear may be adapted for use as either 
azimuth or elevation unit by adjusting gears, 
cams, pins, etc., in compartment to right of elec- 
tric drive motor. 

control system, the system to have considerably 
more output torque than the Ml and M5 sys- 
tems, to have essentially no velocity lag, and to 
be interchangeable with the Ml and M5 sys- 
tems. Both objectives were met at about the 
same time. In view of the superiority of the 
completely new servo system, and its prompt 
adoption by the Army as Oil Gear M3B1 (Fig- 
ures 1 and 2),^^ no appreciable use was made 
by the Army of the indicated means of improv- 
ing the old Ml and M5 systems. 


SPEED REGULATOR FOR MOTORS AND MOTOR GENERATORS 


41 


3 . 4.3 Prototype Design at Westiiigliouse 

The hydraulic servo developed at the Massa- 
chusetts Institute of Technology underwent 
exhaustive tests at the Aberdeen Proving 
Ground and the Antiaircraft Artillery Board, 
and performed well. A project was accordingly 
set up at the Westinghouse Electric and Manu- 
facturing Co.^^ for a production design of these 
servos. In view of the urgent need for the im- 
proved servo the Ordnance Department took 
over direction of the production design and 
procurement of these servos for several bat- 
teries of guns, with a view of standardizing the 
design. A pilot order of 100 units was placed 
with Westinghouse by the Ordnance Depart- 
ment, and on the basis of tests and field trials 
the Ordnance Department changed its produc- 
tion of servos for 37- and 40-mm gun mounts 
to this M3B1 design. The systems employing 
the new servo were termed the remote control 
system M9 for the 37-mm gun mount and the 
remote control system MIO for the 40-mm gun 
mount. 


^ Series of Reports Representative 
of the State of Servo Art (Fall 1943) 

At the conclusion of this work a series of 
Reports to the Services^'^^’^®'^®-^^ were issued 
which represented at that date (December 
1943) a rather comprehensive coverage of the 
linear servo or automatic control theory and 
its use in the design of automatic control 
devices for particular application. In making 
this material available, the division sought to 
bring together papers covering work sponsored 
at the Servomechanisms Laboratory at the 
Massachusetts Institute of Technology by the 
three different sources, namely, contracts with 
the Army Ordnance Department, contracts 
with Section D-2 of NDRC, and independent 
work of the laboratory. Acting as a distributing 
agent with the approval of the sponsors of the 
work, Division 7 sought to disseminate docu- 
ments which, taken together, formed a treatise 
on the current (fall 1943) status of the servo 
art, in particular small servos for application 
to small- and medium-caliber guns. 


35 SPEED REGULATOR FOR MOTORS 
AND MOTOR GENERATORS 


Completing this listing of servomechanisms 
developed under Section 7.3 auspices is a speed 
regulator for an airborne motor generator set 
(Figure 3) developed to satisfy a requirement 




Figure 3. Electronic controlled motor-generator 
set — relay type. Phase sensitive electronic cir- 
cuit for regulating speed of motor-generator, 
contained within box on top of M-G set, is 
shown open. 

for precise gyro speeds in certain gyroscopic 
instruments which will be discussed below. The 
requirement was for electric power of fre- 
quency regulated within a fraction of 1 per 
cent. The section sought to develop a speed 
regulator for a standard Army-Navy airborne 
motor-generator set powered by the usual 
28-volt d-c aircraft supply. Additionally, accu- 
rate regulation of the a-c frequency of the 
generator was required even with a supply volt- 
age variation from 24 to 32 volts. The project 
was undertaken by the Leeds & Northrup Co. 
under contract OEMsr-1292 (Project 81) to 
develop a speed control utilizing a servo loop 
which corrected the d-c motor speed in accord- 


:eDNFipENTIAL 




42 


SERVOMECHANISMS 


ance with deviation error of the generated fre- 
quency from 400 cycles per second. The project 
resulted in the development of a device which 
maintained a frequency within a few thou- 
sandths of 1 per cent at constant d-c voltage, 
and within 1 per cent with supply voltage 
variation between 19 and 36 volts. Several 
motor-generator sets equipped with this speed 
control circuit were turned over to the Navy 
for use in connection with their program on 
the manufacture and tests of the bombsight 
Mark 23.i" 

3 « SEACOAST DATA TRANSMISSION 
SYSTEMS 

® Development at the Bell Telephone 
Laboratories 

In March 1941, the Coast Artillery Board 
‘initiated Project 1207 for the purpose of ob- 
taining a satisfactory solution to the problem 
of continuously transmitting data from base 
end stations to gun data computers. Contract 


the voltage divider system was standardized 
by the Coast Artillery Board and the Ordnance 
Department. 

The system is basically a high-resistance d-c 
Wheatstone bridge, consisting of two poten- 
tiometers, line-balancing rheostats, a power 
source, and a meter indicating unbalance. The 
arrangement is unique in that the potenti- 
ometers have two brushes, and the troublesome 
discontinuity occurring at the ends of the 
potentiometers is avoided by always using the 
center portion of the potentiometers. 

3.6.2 Pilot Model Development at 
Leeds and Northrup Company 

Subsequently, contract OEMsr-404 (Project 
34) was negotiated with the Leeds and North- 
rup Co. to develop a pilot model of the voltage 
divider system. The transmitter assemblies, 
utilizing “aided tracking’’ for tracking the 
target were designed for field mounting on azi- 
muth instrument M1910A1 (Figures 4 and 5) 
and depression position finder Ml, and the re- 



CABLE TO 
JUNCTION BOX 


MOTOR 
FLYWHEEL 
COUPLING - 


HOUSING 

-TRACK" 

SWITCH 

MOTOR "ON-OFF" 
SWITCH 

CONTROLLING 

HANDWHEEL 


INSTRUMENT 

AZIMUTH 

MI9I0AI 


Figure 4. Seacoast data transmission trans- 
mitter, azimuth, M7 showing transmission end 
of resistance bridge telemeter affixed to stand- 
ard telescope receiver. 



OEMsr-177 (Project 20) was negotiated with 
the Bell Telephone Laboratories for this pur- 
pose, and two alternate systems were devel- 
oped: a voltage divider type^° and a permuta- 
tion code type.^^ Both systems gave excellent 
performance as regards both accuracy and 
reliability. However, because of the relative 
simplicity of the former type over the latter. 


Figure 5. Seacoast data transmission M12 and 
M13 receivers. 

ceivers for field mounting on seacoast gun 
data computer MI .22 

Pilot models of two transmitters and two 
receivers were built and in September 1942 
installed at Fort Story, Virginia. After tests 


ge^FlDENTIAL| 


SEACOAST DATA TRANSMISSION SYSTEMS 


43 


on these prototypes by the Coast Artillery 
Board, production orders were placed by the 
Ordnance Department. Toward the end of the 
development contract, the potentiometers were 
modified for transmissions from the SCR-296 
seacoast gunlaying radar. 


3.6.3 Mechanism to Measure the 
Smoothness of Control of 
Aircraft Turrets 

The Bureau of Aeronautics requested the 
development of an aircraft turret smoothness 
tester. This project was placed with the Waugh 
Equipment Co. (Contract OEMsr-1185, Proj- 
ect 75). 


The Bureau of Aeronautics required an 
instrument capable of testing the smoothness 
of aircraft turrets to place acceptance of pro- 
duction turrets on a quantitative basis. A 
primary requirement was to develop an instru- 
ment which did not involve making shaft con- 
nections to the power equipment in the turret, 
so as to facilitate installation of the test equip- 
ment. 

An attempt to use commercially available 
instruments failed because of the very low fre- 
quencies which had to be measured. After 
trying several expedients a capacitance-type 
pickup was designed which responded to very 
small accelerations and low frequencies, and 
which was deemed satisfactory for the purpose 
by the Navy.^^ 


Chapter 4 

PNEUMATIC CONTROLS 


GYROSCOPIC LEAD-COMPUTING 
SIGHTS 

I N AUGUST 1941, Section D-2 distributed a 
report to the Services^ on the fundamental 
dynamics of the gyroscopic lead-computing 
sight.® This paper was the section’s reaction to 
the lack of adequate mathematical material on 
the subject. This study was supplemented in 
August 1942 by a second report^ which sought 
to question how a lead-computing sight should 
be designed to give the highest possible accu- 
racy in its computation of the leads for a 
straight-line target. 

These general studies promptly led to section 
interest in basic improvements of the mechani- 
zation of gyroscopic lead-computing sights. 
Two broad objectives were effected: (1) me- 
chanical refinement of the data-computing ele- 
ments of the instrument, and (2) operational 
improvement by mechanical modification of the 
data presentation means utilized by the sight. 
The former is self evident since improving the 
computation should give better results. The 
latter involves a psychological problem due to 
the nature of the instrument resulting from the 
fact that a lead-computing sight is a “dis- 
turbed” sight.*" 

The first objective involved two types of 
data: the time of flight of the projectile, or 

^ Although the material was specifically directed at 
gyroscopic lead-computing sights, a large part of the 
discussion applied to any sighting mechanism which, 
the gunner directly controlling the gun, sets up be- 
tween the gun and the optical line of sight a lead angle 
which is the product of the time of flight by an angular 
velocity. 

b In a lead-computing sight the gunner exercises 
direct control over the position of the gun and not the 
line of sight. The gunner, by moving his gun, causes 
motion of the line of sight by means of the computing 
mechanism and, eventually, tracks the target with the 
gun displaced in accordance with the proper lead angle. 
Since the gunner has only indirect control over the line 
of sight, a confusing psychological situation may ob- 
tain, for a given motion of the gun in general results 
in a different motion of the line of sight. The gunner’s 
hands, on the gun, attempt to produce a certain result, 
but his eyes see something else happen. For this reason 
a sight of this sort is called a “disturbed” sight. 


range, and the angular rate of the target. 
Range depends upon some target observing 
means, radar range finders, optical range 
finders, or just sheer guess, and can be easily 
introduced into a computing mechanism by a 
hand crank or automatic means. This data can 
be as accurate as desired. Angular rate is ob- 
tained by a suitable mechanism which mea- 
sures the angular rate of the target relative to 
the gun. In order not to measure the angular 
motion of the platform on which the gun is 
mounted it is generally desirable to employ a 
gyro for the angular rate measurement. One 
specific task that Section 7.3 embarked upon 
was to improve the gyroscopic rate of turn 
indicator. 

The second objective was the more elusive 
in that it was desired to present to the gunner 
a line of sight offset by the correct lead angle 
during steady-state tracking, but to decrease 
the time required for the transient solution. To 
decrease the transient solution time constant to 
less than approximately 1.3 times the time of 
flight setting, and to erase false lead angles 
due to slewing on to the target required some 
form of chicanery. 

^ ^ ^ Pneumatic Gyroscopic Lead- 
Computing Sight 

In exploring methods of mechanization at- 
tention was focused on pneumatic controls, 
which had long been used in industrial instru- 
mentation but had not been exploited in the 
fire-control field. The usual method of measure- 
ment of angular rates by gyroscopic rate of 
turn indicators involves constraining a dy- 
namical system, and measuring the forces of 
constraint by s?prings and measuring the force 
by noting the deflection. In order to get a sub- 
stantial response the spring must stretch con- 
siderably to actuate a pointer or some other 
element. In thus stretching the spring, the gyro- 
scope must move, i.e., be imperfectly con- 
strained, and energy is stored in the spring. 




44 


GYROSCOPIC LEAD-COMPUTING SIGHTS 


45 


In an effort to constrain stiffly a gyroscope 
and thus more nearly approach perfect con- 
straint, it was thought that a gyroscope could 
be constrained by forces caused by fluid pres- 
sures which would be varied by slight motions 
of the gyroscopic element, and the fluid pres- 
sures would in turn be used to actuate an in- 
dicator. 


^ Development at the McMath- 

Hulbert Observatory 

Accordingly, contract OEMsr-504 was set up 
at the McMath-Hulbert Observatory (Project 
40) for the purpose of developing a gyroscopic 
lead-computing sight embodying pneumatic ele- 
ments, with primary consideration given to the 
development of a pneumatically constrained 
gyroscopic rate of turn indicator. This purpose 
was achieved to the extent that a rate of turn 
indicator was developed which would measure 
angular rates from several hundred to a tenth 
of a mil per second with a full-scale pressure 
signal of a few pounds per square inch. Fur- 
thermore, the gyroscope was itself constrained 
to within a fraction of a mil angular motion.^ 
However, after the development of the pneu- 
matically constrained gyroscope the exigencies 
of World War II were such that the contractor’s 
efforts were redirected to an urgent program 
for the development of an angular rate bomb- 
sight. Progress Reports 1 to 11 appended to the 
McMath-Hulbert Observatory report^ give an 
account of the lead-computing sight activity. 


Development at the Eastman 
Kodak Company® 

With the shift of emphasis at the observa- 
tory to a program of development of a bomb- 
sight, the Eastman Kodak Co. was asked to 
set up a program under their already active 
contract OEMsr-56 (Project 17) for the devel-^ 

This account omits reference to early exploratory 
work in connection with controlled reticles for lead- 
computing sights. It was hoped that persistence of a 
fluorescent reticle would introduce the necessary 
smoothing of the lead data. For complete details see 
reference 16 of Chapter 2. 


opment of a pneumatic gyroscopic lead-com- 
puting sight.^ This was to take the form of a 
modification of the standard Navy gunsight 
Mark 15 which was being produced at the 
Kodak Co., and designated as the Mark 15-P. 

This problem was also to be interrupted by 
the urgent bombsight program mentioned 
above when the Eastman Kodak Co. was asked 
to design a production version of the bombsight 
developed at the McMath-Hulbert Observatory. 
Before completion of the gunsight problem, 
however, progress pointed clearly to the proba- 
bility of a successful conclusion. However, ter- 
mination of hostilities came before develop- 
ment could be again started and this develop- 
ment must finally be listed under a heading of 
unfinished business at the end of the Eastman 
contract. It is felt that this device, if finished, 
will provide a gunsight embodying several out- 
standing features which are recounted below 
after reviewing the problem in a little more 
detail. 


4.1.4 Navy Gyroscppic Lead- 

Computing Gunsight Mark 15 

As stated above, the Eastman Kodak Co. was 
asked to modify a standard Navy gunsight 
Mark 15. This gunsight is a typical gyroscopic 
lead-computing sight based on the assumption 
that the angle by which the gun must lead the 
line of sight is equal to the angular rate of the 
gun times the time of flight of the shell to the 
target. This assumption is approximate and 
corrections are made by the use of variable 
filter constants, arbitrary functions of range, 
and superelevation corrections. 

It will be recalled that a lead-computing 
sight requires the use of some type of smooth- 
ing circuit in the output, but this filtering in- 
troduces a time lag^ and a “history.” That is, 
all the tracking which goes into slewing the 
sight so that it is on target is remembered by 
the filter circuit. This causes errors for a sub- 
stantial fraction of the course, even after cor- 
rect tracking has been established. 

In the Mark 15 gunsight, two gyroscopes are 
used to measure the elevation and traverse 
components of angular rate. These are coupled 


ioNFIDENTlX 

U - ■ 


46 


PNEUMATIC CONTROLS 


directly to mirrors which “set in” the com- 
ponents of lead angle between the gun and the 
line of sight. Damping is introduced by viscous 
dashpots mounted on the gyroscopes them- 
selves, and a spring with a variable gradient 
exerts a torque on the gyroscope which is in- 
versely proportional to the time of flight set- 
ting. As a result, the time constant of the Alter 
is proportional to the time of flight. 

4.1.5 Prniciples of the Mark 15-P SighP 

Differing from the Mark 15 in several 
respects, the general scheme of the Mark 15-P 
is as follows : A pin joining two slotted cranks 
is used to multiply the precession torque of the 



Figure 1. Lead-computing sight “Mark 15-P.” 

Standard Navy Max'k 15 siR'ht modified for pneumatic 
constrained gyroscope and pneumatic computing elements, 
covers removed, exposing added (shiny) equipment. This 
view shows the extreme compactness of the pneumatic 
computation elements. This one block, only 4%x4%x3% 
inches, contains four pneumatic amplifiers, four variable 
filter elements, four pressure-operated valves, four cali- 
brated pneumatic resistors, and a servo system. 

gyroscope by a factor that represents the time 
of flight (Figure 1). This resultant torque is 
measured as a pneumatic pressure difference 
by a pressure pickup, the pressures are trans- 
mitted through a pair of variable pneumatic 
filters, amplified by a pair of pneumatic am- 
plifiers, and finally made to drive mirrors which 


deflect the line of sight. These operations are 
described more fully in the report.'^ 

^ Advantages of the Mark 15-P System 

The primary advantages of the Mark 15-P 
over the Mark 15 are: 

1. The pneumatic system between the gyro- 
scopes and the mirrors makes it possible to 
turn the computation on or off, and to throw 
away the history at will. Thus this system 
allows the tracker (a) to swing the sight on 
target with fixed optics, (b) to cause the ap- 
proximate lead angle to be applied rapidly to 
the optics, and (c) to introduce the proper time 
constant for that particular range. 

2. Since the pneumatic system does away 
with direct mechanical connections between 
the gyroscopes and the mirrors, it eliminates 
vibration troubles. 

3. A variable time constant makes possible a 
selection of the best relation between time of 
flight and the time constant of the filter, which 
yields a more accurate computation. 

42 DEVELOPMENT OF NAVY 

BOMBSIGHT MARK 23 

The spring of 1942 found German submarine 
activity against Allied shipping so great and 
so effective that ships were being sunk at a 
rate greater than they could be built. This 
situation called for an urgent program on 
countermeasures. Work was accordingly ini- 
tiated with the Franklin Institute, under the 
auspices of Section D-2, and later Section 7.2, 
for development of bombsights suitable for 
directing depth charges against submarines 
from low-flying aircraft. In particular, there 
was developed a bombsight of the conventional 
“depression angle” type, which was standard- 
ized by the Navy as the bombsight Mark 20. 
(See Volume 3.) 

The British Angular Rate Bombsight 
Mark III 

Early in 1943 the British advanced the broad 
idea of determining the bomb release point by 
measurement of the angular rate of the line of 


CONFTDENTL 


DEVELOPMENT OF NAVY BOMBSIGHT MARK 23 


47 


sight. This was in contrast to the conventional 
method of using the angle of the line of sight 
from the horizontal as the release criterion. 
They mechanized this scheme in a device known 
as the British low-level bombsight Mark III.^ 
This sight involved visual comparison of the 



bombsights, angular depression, angular rate, 
and angular acceleration.® (See Volume 3.) 

Development at the McMath- 
Hulbert Observatory 

It was recognized that the new pneumatic 
gyroscopic angular rate indicator (Figure 2) 
developed for use in a lead-computing sight 
could serve as the basis for a bombsight wherein 
the target would be tracked and bombs could be 
released automatically. The McMath-Hulbert 
Observatory project was redirected for the de- 
velopment of a low-altitude angular rate bomb- 
sight. It was felt that not only would the pneu- 
matic bombsight give accurate determination of 
the release point and afford a simple method of 
incorporating smoothing of the tracking data, 
but would also provide simplicity of design 


Figure 2. Pneumatically constrained gyroscope 
for Mark 23 bombsight. 

The sensitive element of the Mark 23 is a 400-cycle 30 
electric gyro. When precessed, the gyrodynamic action 
tends to relieve a spring (upper right side) tensioned in 
accordance with a bombing problem. The gyro is cap- 
tured by a pneumatic potentiometer (upper left), com- 
prising a slotted block rigidly affixed to the housing and a 
cup-pair contained in the slot attached to an arm affixed 
to the gyro. The gyro is thus restricted to angular motions 
of a fraction of an angular mil, the clearance of the cups 
and the block. The pneumatic take-offs are the two tubes 
extending beneath the block. 


relative angular rate of the line of sight and an 
illuminated moving reticle, with manual re- 
lease of bombs by the bombardier at the instant 
of zero relative rate. Because of the necessity 
of judging zero rate and because of the neces- 
sary human reaction time for releasing the 
bombs, it was felt desirable to initiate other 
mechanizations of this principle. Work was 
therefore started at the Franklin Institute 
under Section 7.2 auspices to study the British 
principles and to develop an improved instru- 
ment. The study took the form of a mathe- 
matical evaluation of the various types of 


Figure 3. Bombsight BARB III, Model 2. 

Originally contemplated for hand-held use, this device 
went through many stages of evolution. It finally resulted 
in a device mounted on trunnions but hand-tracked, which 
was standardized by the Navy as the bombsight Mark 23. 

and conservation of weight and space. Work 
was accordingly started at the observatory® on 
the development of an instrument incorpora- 
ting these principles (Figure 3) , which was in- 




48 


PNEUMATIC CONTROLS 


formally designated BARB III (British Angu- 
lar Rate Bombsight) . Parenthetically, the 
mathematical problems involved were intricate, 
since smoothing of the tracking data required 
the solution of certain nonlinear equations ; the 
resources of the Applied Mathematics Panel, 
NDRC, were accordingly drawn upon in their 
solution.' 

In June 1943, a bombsight, hurriedly built 
and embodying the pneumatic features, was 
given preliminary flight tests by the Navy at 
Quonset Point Naval Air Station. These tests 
proved especially gratifying, and the design 
and construction of further improved sights 



Figure 4. Bombsight calibration outfit Mark 2, 
Model 0. 

This device provides a miniature of the horizontal low- 
level bombinR- problem (scale 1/3,000). A Mark 23 bomb- 
sight is shown mounted for test. The lead screw carrys an 
adjustable roller at a constant rate (representing a target) 
and the lever arm resting on the roller angularly swings 
the bombsight as if it were being tracked. The bombing 
problem is preset on the bombsight and the calibration out- 
fit. One of the scales to be set on the calibration outfit 
may be clearly seen on the roller carriage, and the other 
scale is positioned by manipulating the hand crank on the 
extreme right of the outfit. 

incorporating changes recommended by the 
Navy were undertaken at the McMath-Hulbert 
Observatory. 

Certain contributing projects were also 
begun. The first of these was for the develop- 
ment of a bombsight testing engine^ for cali- 
brating the finished sight. This instrument 
afforded a small-scale replica of the actual 
bombing run (scale 1/3,000), the bombsight 
being subjected to ideal angular motion in ac- 
cordance with a preset problem, and the error 


in release time given as a galvanometer deflec- 
tion. This instrument was later standardized by 
the Navy as bombsight calibration outfit Mark 
2, Model 0 (Figure 4). The second item devel- 
oped was a bombrack delay timer.^ This 
afforded a simple means of recording the in- 
terval of time between the bombsight signal 
and the release of the bomb by the bombrack. 



Figure 5. Bombsight Mark 23. 

This pneumatic bombsight is hand-tracked (counter- 
clockwise) while maintaining an illuminated collimated 
reticle of the optical sight on target. Electric power for the 
gyroscope and bomb release mechanism is supplied by the 
armored cable and the case is evacuated through the rub- 
ber hose. The bombing problem is preset by manipulating 
the two knobs on the upper left corner of the case while 
viewing a nomograph through the prismatic (black) lens 
holder on the top of the case. The lamp which illuminated 
the nomograph is exposed. Access plates to the pneumatic 
computing elements are shown in place on the top of the 
case (rectangular plate, abutting front edge of tbe black 
optical sight hood), and on the side (circular plate, lower 
right corner). The button on the top of the right-hand grip 
acts to arm the sight, and the one on the left-hand grip 
actuates the bombardier’s microphone. 

Development at the Eastman 
Kodak Company 

The flight test results of the bombsight 
BARB III led Section 7.3, in anticipation of a 
Navy production program, to request the East- 
man Kodak Co. to divert its facilities and man- 


PNEUMATIC CONTROL ELEMENTS FOR FIRE-CONTROL 


49 


power from the lead-computing sight project BOMBSIGHT MARK 25 

to a program of designing the experimental 

sight for production.® While the performance of the bombsight 

Mark 23 was better than that of the other low- 


Navy Production of the Mark 23 
Bombsight 

While the Eastman Kodak work on a produc- 
tion design was still underway the Navy 
decided to produce the pneumatic angular rate 
bombsight, and standardized it as the bomb- 
sight Mark 23 (Figure 5). The Eastman 
Kodak Co. drawings were turned over to the 
Navy’s contractor, the American Cystoscope 
Co. 

For the reader’s convenience, the tables of 
contents of the McMath-Hulbert Observatory 
and the Eastman Kodak reports are reproduced 
here and serve as an indication of their con- 
tents and scope. 


Final Report on the Mark 23 
Low-Altitude Angular Rate Bombsight, 

McMath-Hulbert Observatory ^ Page 

Introduction 1 

Section I Development of Bombsight Mark 23. __ 6 

Section 11 The Tangent Tracker 16 

Section III Redesign of the Mark 23 32 

Section IV Vibration Testing Program 44 

Section V Theory of Bombsight Mark 23 53 

Section VI Bomb Rack Delay Timer 91 

Appendix I Tables of Tangent Tracker Settings 95 

Appendix II Tables of <//, < and t Calculated for Stand- 
ard Mark 23 Nomographs 121 

Appendix III Tables of Angular Rate for Lighter-than- 

Air Application of Mark 23 145 

Angular Rate Bombsight Mark 23 
Development Dept., Eastman Kodak Co.® 

I General Program 1 

II Principles of Operation 1 

A. Angular Rate Principle 1 


B, Method of Tracking the Sight 

C. Detection of Critical Angular Rate. 

III Model I 

A. Optics 

B. Nomograph 

C. “RC” Smoothing Circuit 

D. Pneumatic Pickup 

E. Erratic Performance 

F. Relay Circuit 

IV Model II 

V Model With Air-Driven Gyroscope 

VI Development of Diaphragm 

A. Diaphragm Ballooning 

B. Effects of Temperature 

C. Design of New Diaphragm 

VII Tests on the Tangent Tracker 

VIII Vibration Tests 

IX Flight Tests 

X Manufacture in Production 


altitude bombsights it was felt that a superior 
mechanization could be developed. In partic- 
ular, it was felt that if the optics could be 
fully stabilized and not subject to the angular 
motions of the aircraft considerable advantage 
could be gained over an instrument which was 
not so stabilized. Also, it was felt that tracking 
by means of a handwheel geared to the optics 
with a reasonable ratio would be preferable to 
direct control of the line of sight ; furthermore, 
it was considered desirable to make the con- 
nection between the handwheel and the optics 
one involving “aided tracking.” 

Work on a stabilized version of BARB 
was started in December 1943, interrupted 
during intense activity on the Mark 23 pro- 
gram between April and December 1944, and 
finally completed in January 1946.® The mecha- 
nization of this sight is more complex than that 
of the Mark 23. It is pertinent to mention that 
Navy tests of this bombsight showed it to be 
capable of solving the low-altitude bombing 
problem with extreme accuracy. This prompted 
the Bureau of Ordnance to standardize the 
sight as the bombsight Mark 25, Mod 0, (Fig- 
ure 6) , and to request Division 7 to transfer all 
apparatus and instruments developed at the 
Eastman Kodak Co. in connection with this 
project to the Naval Ordnance Plant in Indian- 
apolis, Indiana, for continuation of this work. 
A formal Navy report on flight tests of the 
sight is still forthcoming at the time of the 
writing of this report. 

PNEUMATIC CONTROL ELEMENTS 
FOR FIRE-CONTROL APPLICATIONS 

9 Early in 1944 it became apparent to Section 
10 7.3 that a major contribution could be made in 

the field of aircraft instrumentation by exploit- 
13 ing the pneumatic techniques which led to the 
development of the bombsights Mark 23 and 25 

15 and the lead-computing sight Mark 15-P. Ac- 

16 cordingly a broadly defined project was set up 
Jg at the Lawrance Aeronautical Corporation 
19 under contract OEMsr-1366 (Project 82). 


iNFlDENTIAM; 


50 


PNEUMATIC CONTROLS 


This contract was for the development of com- 
pressible fluid controls and stabilizing means 
of general application for fire-control devices. 
In particular, four groups of projects were un- 



r iGURE t). Bombsight Mark 25. 

The housing serves as a rigid support for the gimbal 
frame of a free gyroscope and forms an airtight case about 
the components of the sight. The torque necessary to 
process the gyro in elevation tracking is applied to the 
frame of the gyro by a pneumatic device actuated by the 
hand crank (upper right) , and the bombs are auto- 
matically dropped in accordance with a problem preset by 
manipulation of the knobs bearing the nomograph (lower 
right) . The bombardier views the target through the 
window (front, right) and an optical system carried by the 
gyroscope gimbal. Lateral tracking is given by the knob 
at the top of the case. The protective shell for a pneumatic 
filter is also shown on top of the case. The case is evacu- 
ated through the pet cock on the left side of the case 
(lower left) to supply a working pressure difference for 
the pneumatic mechanisms. The luminous intensity of the 
optical sight is controlled by the black knob on the upper 
left front side of the case. 

dertaken, two of which were supplemental to 
work being done on other contracts. 

Vacuum Regulator for Mark 23 
Bombsight 

In the course of the development of the Mark 
23 bombsight, a standard vacuum regulator 
was adapted to control the vacuum supply to 
the bombsight head. Unfortunately this vacu- 
um control was of the “dump” type which al- 
lowed atmospheric air to flow into the valve 
through a dump port when the vacuum ex- 
ceeded a certain amount. Certain troubles were 
experienced in connection with the Mark 23 
bombsight and it was felt that these troubles 


would be alleviated if a throttle-type regulator 
were used. Since the production sights were 
well under way and since space requirements 
were such that standard throttle-type regula- 
tors could not be used, a throttle-type regulator 
of unusual design was developed of identical 
space factor with the standard regulator.^® 
Several prototypes were made for extended 
tests and samples made available to both the 
Army and Navy. 

^ Elements for Torpedo Control 

In collaboration with Division 6, Section 7.3 
extended its efforts to the field of torpedo con- 
trols under contract OEMsr-1144 (Project 69) 
with the Foxboro Co.^^ The project was set up 
to develop improved steering mechanisms for 
torpedoes, both in azimuth and in depth. How- 
ever, the azimuth steering work was carried 
on at the laboratories of the Division 6 con- 
tractor. The aim of the depth control develop- 
ment was predicated upon an indication that 
the standard depth control was difficult to 
manufacture and marginal in performance, and 
satisfactory depth keeping was obtained only 
when the unit was very carefully made and was 
adjusted by one of a few expert technicians. 
Thus this project sought to develop a depth 
control which would (1) be easier to manu- 
facture, (2) give better performance both as 
regards recovery from launching and during 
the run, (3) be less subject to fore and aft 
accelerations, (4) be preferably lighter in 
weight, and (5) be more rugged than the 
present control. Test runs of this control were 
made in torpedoes at the Newport Torpedo Sta- 
tion, but none of the controls performed satis- 
factorily. In the time available and consistent 
with the other war commitments of the com- 
pany, it was not possible for the contractor to 
complete the development of an improved depth 
unit. Much progress was made however, and 
inasmuch as the general features of the design 
appear sound, it is expected that the Navy will 
continue this work. 

In furtherance of the work at the Foxboro 
Co. a project was set up under the Lawrance 
Aeronautical Corporation contract for the im- 
provement and development of a depth engine 
for torpedoes. This depth engine was not a 



THEORETICAL ASPECTS AND DESIGN CRITERIA 


51 


sufficient improvement over the present stand- 
ard device to warrant a change, and the work 
accordingly was not extended beyond the devel- 
opmental stage. 

A third torpedo component developed at the 
Lawrance Aeronautical Corporation was a 
500-psi torpedo regulating valve, which was the 
pressure reducing valve for the 3,000-lb air 
supply. The objective was to design a regulator 
valve for torpedoes for easy manufacture and 
for stability of operation without expert ad- 
justment. The regulator from a Mark 15 tor- 
pedo was modified to this end and laboratory 
tests indicate the design to be satisfactory. The 
design and test information and a model were 
turned over to the Navy for use in guiding the 
subsequent design for torpedo pressure regu- 
lators.^^ 

Stabilization of Aerial Cameras 


ume 3.) Units developed under the contract 
for this purpose were (1) an absolute pressure 
regulator, reference pressure regulator, and 
relief valve, (2) twin variable pneumatic re- 
sistors, (3) a servo operator for use on the 
PUSS gyro, (4) a solenoid valve of special 
design, and (5) vane motors for the optics of 
PUSS.i® These components with complete 
design and test information were turned over 
to the Franklin Institute for use in carrying 
the PUSS project to completion under a direct 
Navy contract (Navy contract NOrd-9644). 

Also in connection with the PUSS project, 
work was done under contract OEMsr-56 at 
the Eastman Kodak Co. (Project 17) to design 
and construct a pneumatically constrained 
gyro. A model gyro was turned over to the 
Franklin Institute for incorporation in the sys- 
tem after laboratory tests indicated it was 
satisfactory for the application.^® 


In conjunction with a Division 16 program 
for the development of cameras and camera 
stabilizers, work was done under the Lawrance 
contract on the stabilization of cameras by 
pneumatic means. Initial work dealt with a 
camera stabilizer suitable for reconnaissance 
photography. A camera stabilizer for mapping 
application was undertaken after the explora- 
tory phase was completed. The result was a 
laboratory model which provided excellent 
stabilization of the camera against motion of 
the platform (simulated aircraft angular mo- 
tion) and which provided for maintaining the 
camera axis at the effective vertical within 
close limits.^^ 

This equipment was demonstrated to the 
representatives of the Photographic Labora- 
tory of Wright Field under whose auspices this 
project was initiated (AC-76), and the Army 
has placed a contract calling for further work 
with fhe engineers responsible for this devel- 
opment. 

4.4.4 Pilots Universal Sighting System 
[PUSS] 

In collaboration with Section 7.2, work was 
done under the Lawrance Aeronautical Cor- 
poration contract on pneumatic components 
for a pneumatic version of PUSS.^® (See Vol- 


Other Pneumatic Circuit Components 

Four miscellaneous items were also made 
under the Lawrance contract, two of which 
were exploratory models of devices which were 
redesigned as PUSS components: a pressure 
amplifier,^^ and a pneumatic resistor.^® The 
other two items developed were under an in- 
formal request by the Bureau of Ordnance for 
a dive angle indicator embodying the cup-con- 
straining features used in connection with the 
Mark 23 and the torpedo developments.^^ ggy. 
eral dive angle indicators were assembled and 
turned over to the Bureau of Ordnance for use 
in its development programs, particularly with 
regard to toss bombing. 

At the Eastman Kodak Co. a pneumatic 
phase-controlled speed-regulating device for air 
gyros^® was developed for the purpose of 
adapting this unit to air-driven rate gyros in 
applications when precise gyro speeds are re- 
quired. This speed control device employed a 
vibrating reed as the frequency standard, and 
gave excellent speed regulation. 

THEORETICAL ASPECTS AND 
DESIGN CRITERIA 

During the course of development of the 
various pneumatic instruments discussed above. 



52 


PNEUMATIC CONTROLS 


Section 7.3 turned to the basic problem of their 
theory of operation. In particular, a problem 
was initiated under the Lawrance Aeronautical 
contract to study the components of the sys- 
tems developed and to compile the design 
criteria which existed only on odd charts and 
in the notebooks of the engineers working on 
the projects. 


Literature on Pneumatic 
Instrumentation 

Pneumatic instruments are common in in- 
dustry, where pressure, fluid flow, and similar 
physical quantities must be controlled. The 
literature covering the design and theory of 
these instruments is quite complete.^o ^i Certain 
aircraft instruments (rate of climb indicators, 
airspeed indicators, altimeters, etc.) are also 
common pneumatic devices. These have been 
analyzed and make up a considerable body of 
literature. 22 - 2 ^ It may be of interest to recall 
that many examples of both these classes are 
to be found in the patent literature. Seldom, 
however, does there appear a reference in this 
fleld to means of accomplishing a given task 
pneumatically in analogy with the usual way of 
performing the job in other fields, say, elec- 
trically or mechanically^® although the analogs 
are old and well known.^^-^i 

Indeed, one must turn to the field of acoustics 
for an introduction to a comprehensive treat- 
ment of pneumatic theory. The modern 
treatises, firmly rooted in the classic ex- 
positions of Rayleigh®® and Lamb,®^ supply 
valuable source material for a mathematical 
analysis of pneumatic instrumentation phe- 
nomenon. Here we find the notions of fluid 
pressures as voltage, fluid flow as current, the 
capillary as a resistance element which serve 
as a basis of analysis of systems using these 
quantities and/or elements as components. 


Mathematical Studies 

Lacking handbooks and similar guides which 
give in tabular form particular values for these 
various quantities as a function of dimensions 


and ambient conditions. Section 7.3 was obliged 
to draw up design charts for this purpose. This 
required recourse to the field of acoustics and 
the works referred to above. Subsequently 
these charts were expanded and a report,®^ 
issued under the Lawrance contract, tabulated 
these data in a form similar to Massa’s acoustic 
design charts.®® Mathematical work of an ad- 
vanced nature on particular pneumatic com- 
ponents was begun during the life of the con- 
tract, but was never concluded because of the 
termination of World War II. 37.38 


Comparison Study of Gyroscopes 
and Gyroscope Substitutes 

During World War II two gyroscope substi- 
tutes were proposed. One was a “fluid gyro”®® 
and another made use of a vibrating reed.'^® 
The fluid gyro was proposed by the Columbia 
University Division of War Research at the 
U. S. Navy Underwater Sound Laboratory, 
New London, Connecticut. This was fashioned 
as a case with a vaned passage of axial sym- 
metry through which a fluid current is directed. 
The case being subjected to a torque about the 
axis imparts a moment of momentum to the 
fluid, and the angular momentum of the fluid 
is subsequently used as an indication of the 
angular velocity given the case with respect to 
a body pendulus with respect to the axis. 

The vibrating reed made use of the fact that 
a rod anchored at one end and maintained in a 
state of vibration about its anchorage tends to 
maintain its vibration in a frame which is fixed 
in space. 

A series of studies were carried out under 
contract OEMsr-268 with the Barber-Colman 
Co. to compare these two devices with an ordi- 
nary gyroscope of typical parameters and the 
results form appendices to the final report 
under that project.^^ The conclusions were sum- 
marized as follows: 

1 . When stabilization about one axis only is 
required, it is difficult to state which instru- 
ment will have the more accurate dynamic 
response. For sinusoidal forcing accelerations 
the standard gyro proved to be considerably 
more accurate. For the other types of motion 


THEORETICAL ASPECTS AND DESIGN CRITERIA 


53 


considered, it is impossible to say in general 
that either will be more accurate than the 
other. 

2. When stabilization about two axes is re- 
quired, two fluid gyros should prove more accu- 


rate than a single standard gyro for all types 
of forcing accelerations considered. 

3. The fluid gyro should have considerably 
smaller static friction error than the standard 
gyro. 


CONFIDENTIAL 


Chapter 5 

MATHEMATICAL ANALYSIS OF FIRE-CONTROL PROBLEMS 


51 INTRODUCTION 

F our groups of projects to be summarized 
below are concerned with contributions of 
the Mathematical Analysis Section, Section 7.5, 
to the fire-control research program of NDRC. 
Several of these projects were begun under 
Section D-2 auspices and came to fruition be- 
fore the reorganization of NDRC. Other proj- 
ects were, for reasons of administration ex- 
pediency, transferred to the Applied Mathe- 
matics Panel at the time of its organization. 

Because the Chief of Section D-2 and of Sec- 
tion 7.5 became the Chief of the Applied 
Mathematics Panel, and because the Division 
7 activities in mathematical analysis, as a 
special case, were incorporated in the more 
general applied mathematics picture of the 
panel, the present account of mathematical an- 
alysis of fire-control problems must necessarily 
be partial. For a more integrated and compre- 
hensive account, the reader should consult the 
Summary Technical Report of the Applied 
Mathematics Panel. 

Ten projects of Division 7 were in the prov- 
ince of mathematical analysis. Two dealt with 
statistical prediction, three with mathematical 
aids for computation, two may be designated 
as miscellaneous, and three were transferred to 
the Applied Mathematics Panel. These will be 
taken up in the order given. 


5 2 STATISTICAL THEORY OF 

PREDICTION 

It is essential, in the design of any anti- 
aircraft or other director, to have a device 
which calculates some estimate of the future 
coordinates of the target, basing this calcula- 
tion on a knowledge of the present and the past 
values of the coordinates and the derivatives 
thereof. Two main types of methods are of 
particular significance for the calculating of 


future values. One, which is the method used 
in the past and which will be called here the 
geometrical method, rests upon certain as- 
sumptions as to the geometrical character of 
the path of the target (say the assumption that 
it moves on a straight line), and the future 
position which would then result is computed 
on the basis of this assumption. 

The actual computation of future or pre- 
dicted position is rather simple. The method 
is inherently capable of perfect precision, the 
actual practical accuracy being limited by the 
accuracy (which may in some cases be excel- 
lent, and may in others be low) with which 
one knows the parameters (say present posi- 
tion and velocities) which determine the geo- 
metrical path in question ; and by the accuracy 
(which may again be high or very low) with 
which the actual flight path of the target con- 
forms to the assumption made as to its geo- 
metric character. 

During World War II various generaliza- 
tions of this basic geometrical method were 
proposed. These generalizations involving some 
type of extrapolation to future value are based 
upon present values and some range of past 
values. In this extrapolation procedure the pre- 
diction is usually expressed as the sum of 
terms of decreasing importance; and one uses, 
in the actual mechanized solution, as many 
terms as prove practicable. The calculation it- 
self therefore usually involves approximation. 

This extrapolation procedure is merely a 
generalization of the older and more familiar 
form of the geometrical method ; for any extra- 
polation formula adopted constitutes an as- 
sumption as to the geometrical character of 
the path. The extrapolation method at first 
sight appears to differ in that it makes explicit 
use of past data as well as present data. But 
this distinction is an illusion, for the geo- 
metrical method, even in the simple case of 
straight-line motion, necessarily uses past data 
to compute velocities. Indeed, because of the 


^^ ^NFIDENTIAljB 


54 


STATISTICAL THEORY OF PREDICTION 


55 


errors inherent in target motion, observing of 
target motion, and computing, it is necessary 
to use quite a range of values in order to com- 
pute a usefully approximate value for an 
“instantaneous” velocity. The extrapolation 
method recognizes this situation explicitly, and 
uses a range of past data to help compute the 
geometrical parameters of the target path. 

^ Statistical Method of Prediction 
(Projects 6 and 29) 

Under contract NDCrc-83 (Project 6) a 
mathematical theory of a second and more 
general type of a prediction, which may be 
termed the statistical method, was worked 
out.“ The statistical method, as a preliminary 
step, utilizes the older geometrical method to 
make an approximation to the prediction; but 
by comparing the actual present values with 
the first-order predictions made in the past 
and culminating at the present, one has avail- 
able as present data the amounts of f{t) by 
which the simple geometric or extrapolation 
method fails. This deviation f(t) is assumed 
to have two additive components, the first of 
which, /i(0 (the “signal”), is principally due 
to the fact that the actual motion of the target 
does not conform to the assumptions made in 
the simple prediction ; and the second of which, 
fzit) (the “noise”), is due to errors of various 
sorts. 

Wiener worked out mathematical methods 
which statistically examine the past behavior 
of the signal f^(t) and produce, at the present 
time t, the “best possible” guess as to what 
value this signal /i(f) will have at a future 
time. The phrase “best possible” has a definite 
meaning in the least square sense. The theory 
also included a determination of the character- 
istics of electrical networks which are capable 
of carrying out the necessary operations. It is 
necessary (and possible) to design the net- 
works in a general way, leaving certain im- 
portant characteristics adjustable. To set the 
network so as to do the best job of statistical 
predicting of the future for a given target, one 

" By Norbert Wiener of the Massachusetts Institute 
of Technology. 


sets the adjustable characteristics of the net- 
work in accordance with certain statistical 
parameters which characterize the past motion, 
which describe, so to speak, the “habits” of the 
motion. These statistical parameters are them- 
selves determined by the machine ; and the pre- 
dicting network may be so flexibly adjustable 
that it is able to handle motions of widely dif- 
fering statistical characteristics. 

The preceding paragraph refers only to the 
signal component f^{t). The theory also pro- 
vided means for the best possible filtering of 
the total quantity f{t) into its signal compo- 
nent /i(t) and its noise component ^(t). 

The successful performance of a simplified 
laboratory model of a statistical predicting 
network designed in accordance with the 
theory was demonstrated. In order to carry 
out such a demonstration, it was necessary to 
have a suitable type of input signal to predict. 
There was constructed, therefore, a device in 
which one attempted to track a sinusoidally 
moving spot of light by means of a second spot 
over which the operator had a loosely coupled 
control through a somewhat complicated mech- 
anism involving large inertia, spring coupling, 
damping, etc. The “error” between the track- 
ing spot and the tracked spot is then taken as 
the statistical input to the predicting circuit. 

The circuit demonstrated its capacity to pre- 
dict on such a signal over an interval of as long 
as 2 seconds. For a 1-second prediction the 
correspondence between the predicted value 
and the ultimately realized value was astonish- 
ingly close. 

Whether or not this interesting procedure 
has important application to long-range anti- 
aircraft fire control depends on the following 
considerations. Taking only one coordinate of 
the target plane, the input to the predictor con- 
tains (1) “flight errors” corresponding to actu- 
al movements of the airplane due to bumpy air 
or to the pilot himself, (2) long-period track- 
ing errors, (3) short-period tracking errors, 
and (4) “noise” or other small errors due to 
miscellaneous causes. The question is: Are 
these various errors sufficiently separated in 
frequency so that Wiener’s network can pre- 
dict the flight errors and at the same time filter 


s FIDENTJAL j 


56 


MATHEMATICAL ANALYSIS OF FIRE-CONTROL PROBLEMS 


out (or at least not predict) the long-period 
and short-period tracking errors ? Moreover, is 
there some region in which the noise level is 
low enough to tolerate the high gain of his cir- 
cuit and can the prediction be made for a length 
of time which is practicable? 

After an extensive series of visits to the 
commercial laboratories, government and Ser- 
vice organizations, and individuals to obtain 
information concerning antiaircraft tracking 
errors, frequency characteristics of input sig- 
nals, flight errors, etc., sufficient information 
was obtained to answer the questions raised 
in the previous paragraph, and a report on this 
subject was submitted.^ In it is compared the 
effectiveness of antiaircraft fire using a fixed 
memory point predictor, a predictor using a 
memory point trailing 10 seconds behind the 
present, and a statistical predictor for certain 
actual target aircraft courses at Camp Davis. 
For the actual courses studied, the second 
method is notably better than the first, but 
the third is no better than the second. 

Thus it seems at present doubtful indeed 
whether this beautiful mathematical theory 
has direct practical application to the problem 
of predicting the future position of aircraft 
targets. It has produced one valuable result 
by providing a reference or standard of com- 
parison in terms of which the present predic- 
tion methods can be rated. Thus the fact that 
the very fundamental statistical method yields 
no significant gain in accuracy over present 
methods is an important guide for future 
developments. 

It seems highly probable, however, that the 
theory will have applications to various phases 
of the general fire-control problem of filtering 
out signal from noise, and of analyzing the 
smoothing-prediction problem under various 
circumstances. It also seems inevitable that 
this general and powerful analysis will have 
important applications to other statistical 
problems. 

The project was terminated in February 
1943 on the basis of the above stated evidence 
of the improvement which the statistical 
method can promise, and on the basis of Ser- 
vice advice concerning the general importance 
of curved-flight predictors. 


Report on the Extrapolation, Inter- 
polation, and Smoothing of Stationary 
Time Series with Engineering 
Applications 

As a by-product of the work in Project 6, 
certain general mathematical methods were 
developed which were believed to be useful in 
other fields, particularly in statistical work 
and in the design of electrical filter circuits. 
Copies of a report^ covering these phases were 
distributed as Report to the Services No. 19. 
This report was the subject of considerable 
discussion.^’^'^ 


5 3 MECHANICAL AIDS TO 

COMPUTATION 

In several instances during World War II, 
routine numerical computation became a bottle- 
neck in the carrying out of fire-control testing 
programs. This focused attention on the pos- 
sibility of carrying out such computations by 
means of automatic calculating devices. 

The first project to which automatic compu- 
tation was applied was the preparation of data 
for an antiaircraft testing device called the 
tape dynamic tester which has been reported 
upon by Section 7.1. For the operation of this 
instrument, it was necessary to have a long 
paper tape in which is punched a series of holes 
representing numerical data in an arbitary 
code scheme, these data being values of one of 
the coordinates of an aerial target. 

In typical cases one computes, by hand, the 
three coordinates (azimuth, elevation, range) of 
a fictitious target following a certain desired 
test course, computing these three data for 
points, say, 1 second apart on the course. The 
tape dynamic tester, however, requires these 
data at intervals of only %o of ^ second. Thus 
a computing device was required which would 
accept the 1-second interval data, and compute 
from these, using interpolation formulas in- 
volving third differences, 19 intermediate val- 
ues between every two of the original values. 

Since the test courses involve six functions 
and average about 180 seconds in length, each 
course involves roughly 20,000 interpolations. 


MECHANICAL AIDS TO COMPUTATION 


57 


In the life of the project over 60 courses in all 
were produced, so that well over 1,000,000 in- 
terpolations were called for. This represents 
the equivalent of about 3 years of a skilled 
computer’s time. In addition, the million in- 
terpolated numbers were automatically trans- 
lated into an arbitrary code and holes punched 
in tape, no errors or corrections being allowed. 
It is difficult to estimate the time required to do 
this latter step manually, since even skilled 
operators found difficulty in going through a 
long tape without errors. 

Relay Interpolator (Project 70) 

The foregoing considerations made it desir- 
able to construct a machine capable of inter- 
polating, translating into code, and punching 
the tapes with a minimum of supervision. Con- 
sequently, a device was designed by Division 7 
in which the required tasks would be done by 
telephone relays in accordance with formulas 
represented by codes punched in a control tape. 
The schematic and some of the detailed circuits 
were turned over to the Bell Telephone Labora- 
tories which agreed to build the relay inter- 
polator embodying the schematic ; and con- 
tract OEMsr-1160 was set up for this purpose 
with the Western Electric Co., dated July 1, 
1943. In 10 weeks the machine was in opera- 
tion, and continued to be used up to the ter- 
mination of World War II. In addition to 
preparing dynamic tester tapes, the relay in- 
terpolator has been found useful in many 
other interpolation problems, and in fact in a 
rather surprising range of computing prob- 
lems, whether or not interpolation is involved. 

About September 1, 1945, the relay interpo- 
lator was moved to the Naval Research Labo- 
ratory at Anacostia, where it is now in opera- 
tion. Several reports have been written on the 
theory and operation of the interpolator and 
have been distributed by Division 7 and the 
Applied Mathematics Panel.®-® 

Ballistic Computer (Project 74) 

A similar situation with regard to computa- 
tion arose at the Anti-Aircraft Board. The 
amount of computation to be done in working 


up test data far exceeded the manpower avail- 
able. In view of the successful application of 
relays to the interpolation problem just de- 
scribed, the division decided to use similar 
means to assist the board. 

Again, schematics were prepared by the 
division. Several new features were required 
to solve the problems presented. One of these 
was means for multiplying numbers. Another 
was means for storing ballistic tables. In place 
of the circuit for multiplying by repeated ad- 
dition and shifting as suggested by the division. 
Bell Telephone Laboratories, under contract 
OEMsr-1236, chose to use a scheme devised by 
themselves which employs a multiplication 
table stored on relays. The storage of tabular 
values was carried out, as suggested by Di- 
vision 7, by punching the data and interpola- 
tion coefficients on tape, together with the 
arguments. A hunting circuit causes the tape 
transmitter to move the tape forward or back- 
ward until the desired argument is reached, 
and the data and interpolation coefficients are 
read off. 

The contract was set up November 12, 1943, 
and on the night of May 12, 1944, the Anti- 
Aircraft Artillery Board [AAAB] computer, 
constructed in accordance with these plans, 
completed its first unattended run of ballistic 
computations. 

The machine is capable of almost any se- 
quence of computations involving multiplica- 
tion, division, addition, and subtraction of 5- 
digit numbers. It will store 10 such numbers 
simultaneously, and will hold and search 
through “tables” of functions, each capable of 
storing about 10,000 digits. The problem data 
are introduced on two punched tapes, one of 
which may also be searched. There is practi- 
cally no limit to the number of steps (addition, 
searches, etc.) which can be combined into a 
problem. Ordinary ballistic problems involve 
200 to 300 such steps. 

In a typical problem, the operator receives 
the results of a test on an antiaircraft director 
in the form of a table which shows, usually at 
1-second intervals, the three coordinates, range, 
azimuth, and elevation angle, for the target, 
and the three gun orders, angle of train, quad- 
rant elevation, and fuze setting, as calculated 


mi'IDENTIAffl 



58 


MATHEMATICAL ANALYSIS OF FIRE-CONTROL PROBLEMS 


by the predictor. It is required to compute the 
error in this prediction. 

With the AAAB computer, a girl transcribes 
the target coordinates and the corresponding 
times on punched tape, and the gun orders on 
another tape. Ordinarily, several courses of 
200 or 300 sets of data will be transcribed on 
each tape. The two tapes are then placed in 
the computer, in which there are tapes carry- 
ing ballistic tables and a tape on which are 
formulas to be used in the computation. The 
operator pushes the start key and leaves it to 
the machine to do the rest. 

The computing machine reads the target ele- 
vation and range at the first recorded instant. 
It hunts the entry on the ballistic table tape 
which is nearest this pair of arguments, and 
interpolates by a quadratic formula in the two 
variables for the time of flight of a shell to the 
target. Next, it reads the time at which the 
data was recorded, and subtracts the time of 
flight to get the correct firing time. 

The computer now needs to know what data 
the director calculated at the firing time. It re- 
fers to the gun order tape and selects the four 
closest instants, from which it interpolates by 
a cubic formula for the gun orders transmitted 
by the director at the firing time. As it obtains 
the interpolated value for each gun order — 
fuze, quadrant elevation, and angle of train — 
it interpolates in the ballistic table to get the 
correct value, and prints the difference between 
correct and actual gun orders in the proper 
column of a result sheet. 

The time required by an operator to carry 
out these computations with the help of the 
usual commercial calculators is about 40 min- 
utes per set of data. The AAAB computer does 
the same job in 150 seconds, and works prac- 
tically 24 hours per day. Thus it does the work 
of about 50 human operators, with the help of 
two girls to transcribe data and one part-time 
maintenance man. If the cost of the machine, 
about $120,000, is amortized in 3 years (a very 
conservative figure, since little replacement will 
be required in 5 or 10 years), then the cost of 
the man-year of work is well under $1,000 or 
about Vs the cost of manual computation. A 
more important consideration, however, is that 


the overall delay in obtaining the results of 
tests on equipment was greatly reduced. 

As a direct outgrowth of the NDRC relay 
computer development and because of their 
high order of reliability (experience at Fort 
Bliss having shown that about one trouble per 
week would occur on the average, and would 
require i/o to 1 hour to find and clear), three 
other computers have been made. The first of 
these is practically a duplicate of the AAAB 
computer which is running at the Naval Re- 
search Laboratory. Two much larger machines 
have been designed in consultation with Section 
7.1 and are being built for the National Advis- 
ory Committee on Aeronautics and for the 
Aberdeen Proving Ground. 


® 33 Investigation of the Differential 
Analyzer (Project 62) 

During World War II, the Moore School of 
Electrical Engineering at the University of 
Pennsylvania operated its differential analyzer 
on a full-time basis in the solution of the dif- 
ferential equations of exterior ballistics under 
direct contract with the Aberdeen Proving 
Ground. Because of the large amount of work 
to be done both at Aberdeen and at the Uni- 
versity of Pennsylvania, it proved desirable to 
improve the efficiency of the machines in use. 
It was hoped that, as a result, manufacturing 
designs and specifications would be prepared 
for replacement units which could then be pro- 
cured directly by the Ordnance Department 
and the University of Pennsylvania. 

A project was thus initiated at the Univer- 
sity of Pennsylvania under contract OEMsr- 
856 and the following program was planned: 

1. A study was to be made of the slip oc- 
curring in integrators, its sources and means 
of reducing it in order to operate the units at 
higher speed. Experimental equipment was 
built for measuring integrator slip. 

2. A study was to be undertaken leading to 
the immediate application of available and po- 
tentially useful torque amplifiers. The ones 
used in the analyzer, although satisfactory 
under ordinary conditions, were not able to 



MISCELLANEOUS PROBLEMS 


59 


perform under conditions of full-time employ- 
ment. The maintenance required was excessive 
and the accuracy of the integrators impaired. 

3. The preceding two items were hopefully 
intended to raise the operating level of the 
machine by a factor of two or three. If and 
when this was done, it would be necessary to 
take into account the effect of increased speed 
on other units. A new counter tripping unit 
would be required to provide compensation for 
the time lag in the counter system. Also, sev- 
eral special input tables, then in use and oper- 
ating at the lower speed level, would have to 
be refined in order to operate successfully at 
higher speeds. 

4. The development of special data-transfer 
equipment to permit use of improved ballistic 
methods was to be undertaken. This item was 
essential to the efficient utilization of otherwise 
unused integrators. 

During the work doubt arose as to the ac- 
curacy of the experimental setup for measur- 
ing integrator slip. It was therefore decided 
to review this aspect of the program very care- 
fully to determine what useful results, if any, 
could be obtained with the existing setup; 
what problems relating to integrator slip ap- 
peared to be most important ; and what experi- 
mental setup would apparently be necessary 
to solve those problems. 

Under item 2, the then available types of 
torque amplifiers, including a Polaroid type 
developed by the General Electric Co., a modi- 
fied Polaroid type, a Maxson hydraulic unit, 
and the torque amplifier used in the M5 anti- 
aircraft predictor, were tested. None of them 
was found satisfactory for the particular ap- 
plication. In some instances the torque output 
was insufficient; in some instances the maxi- 
mum speed was too low ; in some instances the 
input torque was too great; and in some in- 
stances the torque amplifiers were unstable 
when connected in series. 

The contract was thus extended in time, and 
modestly supplemented in funds, for the pur- 
pose of finishing up those aspects of the inte- 
grator slip problem which could be handled 
with the experimental setup, and to make pos- 
sible the completion of the survey of torque 


amplifiers for use in differential analyzers. 
Changes were made (chiefly involving an im- 
proved optical system) in the Polaroid type of 
torque amplifier referred to above. At the same 
time progress was made in improving the bands 
and strings for the Niemann type of torque 
amplifier now used on the differential analyzers 
at the Massachusetts Institute of Technology 
(that is, on the older model), at Aberdeen, at 
the University of Pennsylvania, and elsewhere. 

The modified torque amplifier aspect of the 
study was brought to a successful conclusion, 
and a contract with the Army negotiated under 
which eight servos of the new type were con- 
structed to carry out practical tests on the dif- 
ferential analyzer. With exhaustion of contract 
funds, certain interim work was carried on by 
the University of Pennsylvania, pending the 
arrangements for the new contract with the 
Ordnance Department. This interim work was 
devoted to developing a curve-following mech- 
anism capable of speeds comparable with that 
of the amplidyne-Polaroid servomechanisms 
previously developed. On tests, the photocell- 
controlled mechanism operated satisfactorily 
when following the edge of a line % inch wide. 

Further details are available in the final re- 
port under the contract.^^ 


MISCELLANEOUS PROBLEMS 

The services of two mathematicians'" were 
made available by placing contracts with the 
Princeton University (contract NDCrc-105) 
and the University of Wisconsin (contract 
NDCrc-116). Work on the latter was inter- 
rupted, and the contract was terminated before 
definitive results were obtained. Work on the 
former resulted in five studies: 

Some Experimental Results on the Deflection 
Mechanisms . This is a detailed study of the 
accuracy and stability of a deflection mech- 
anism produced by the fire-control design group 
at the Frankford Arsenal. In connection with 
this study extensive use was made of the differ- 
ential analyzer at the Massachusetts Institute of 
Technology. 

^ C. E. Shannon and I. S. Sokolnikoff. 


NFinKNTIAn^ 


60 


MATHEMATICAL ANALYSIS OF FIRE-CONTROL PROBLEMS 


Backlash in Overdamped Systems.^^ The pre- 
ceding report^^ showed that backlash could 
cause sustained oscillation in a second-order 
mechanical system, provided that it was less 
than critically damped. This paper attacks the 
same problem for overdamped systems. 

The Theory of Linear Differential and 
Smoothing Operators.^^ This is a general study 
of predicting and smoothing mechanisms. 

A Height Data Smoothing Mechanism.^^ 
This is an analytical study of a particular 
mechanism which might be used for the 
smoothing of height data. 

The Theory and Design of Linear Differen- 
tial Equation Machines.^^ This paper gives a 
general mathematical theory which permits the 
rapid analysis of mechanical computing cir- 
cuits and the design of mechanical computing 
circuits having desired characteristics. 


55 PROJECTS TRANSFERRED TO THE 
APPLIED MATHEMATICS PANEL 


® ® ^ Statistics of Train Bombing 
(Project 23) 

This project was begun in July 1941 under 
contract OEMsr-817, originally administered 
by Princeton University. Subsequently the 
work was expanded to include a contract with 
the University of California, and the Prince- 
ton contract was expanded and shifted to Co- 
lumbia University. The project originated in 
a request for the design of a bombardier’s cal- 
culator, but subsequently developed into a broad 
study of the statistics of train bombing. 

Basic tables for the probability of at least 
one hit on various rectangular targets were cal- 
culated, showing the way in which this proba- 
bility depends upon the number of bombs in 
the train, the bomb spacing, the angle of ap- 
proach, the size and proportion of the target, 
and the magnitude of the aiming and disper- 
sion errors. 

Extensive studies were carried out to extend 
these results to produce a general theory of 
multiple hits on multiple targets. This theory 
analyzes the way in which missions should be 


planned and attacks carried out in order to 
maximize the number of cases in which at least 
k hits = l to 5) will be obtained. 

Studies were made to determine whether or 
not these theories could be usefully applied 
under circumstances where (as must always be 
the case) the aiming errors are not actually 
known. The theory of multiple hits was checked 
by a study of a train-bombing experiment at 
Eglin Field, and also by an experiment carried 
out for this purpose on a bombsight trainer. 

Studies were under way at the date of trans- 
fer of the project to the Applied Mathematics 
Panel to determine the optimum type of attack 
on maneuvering targets. Five reports were is- 
sued under this project.^^’^^ 

Computations (Project 39) 

In connection with a variety of projects and 
plans, the division found it increasingly neces- 
sary to have computations performed. These 
were both of routine and of highly specialized 
character. In each case it was most efficient 
to have the actual work performed at the par- 
ticular place best equipped to handle it. This 
project was begun under the auspices of the 
Franklin Institute under contract OEMsr-444, 
enabling the division to carry out computations 
whenever the necessity arose by requesting the 
Franklin Institute to procure the work from 
the most effective agency. 

The contract proved exceedingly useful. 
Under it were supported a study of the differ- 
ential analyzer-ballistics problem, a study of 
the fragmentation-damage problem, a study of 
scatter bombing, etc. Since formal reports 
were not required under the terms of the con- 
tract none was submitted. The reader should 
therefore turn to the Summary Technical Re- 
port of the Applied Mathematics Panel for a 
bibliography. 

* 5.3 Warfare Analysis (Project 47) 

As a result of a conference held by the Fire- 
Control Division with representatives of the 
Naval Bureau of Ordnance, the Naval Bureau 
of Aeronautics, the Office of the Coordinator 
of Research of the Navy, and the Army Air 


X^FIDEN1IAI|: 


PROJECTS TRANSFERRED TO AMP 


61 


Corps, there was planned a general program 
for a series of probability and statistical stud- 
ies of plane-to-plane fire. It was hoped that 
such a stud 3 % by evaluating the influence of the 
various factors on the overall effectiveness of 
plane-to-plane fire, would give some practical 
information on certain problems of design of 
airborne fire-control systems. Accordingly con- 
tract OEMsr-618 was negotiated with Colum- 
bia University for this purpose. 

At a later conference with Service personnel 
it was agreed that the group could most use- 
fully first attack the problem of estimating, 
through probability considerations, the com- 
parative effectiveness of various mixed batter- 
ies for a fixed gun fighter attacking a bomber. 
Those connected with the project visited the 
AAU at Norfolk and had firsthand contacts 
with aircraft fire-control equipment. 

Early in 1943 the project was expanded in 
personnel and broadened in scope so that it 
could undertake any studies under the general 


title of Air Warfare Analysis. The group was 
then composed of about fifteen technically 
trained people. Among the studies that were 
undertaken as a result of direct requests from 
the Services or other NDRC divisions were: 
statistical acceptance tests for bombsights ; an- 
alysis of a dive bombsight; estimate of addi- 
tional risks to a bomber due to extensions of 
the straight bombing run; probability of dam- 
age to a dive bomber; optimum ammunition 
for air combat, and counter-evasion measures 
for aerial torpedoing; problems bearing upon 
the statistical aspects of the testing of certain 
naval antiaircraft fire-control equipment; and 
problems relating to the broad aspects of prob- 
ability of damage to aircraft through antiair- 
craft fire, plane vulnerability, optimum inter- 
relations of aiming errors and gun dispersions, 
etc. 

For a detailed account of this work the 
reader is referred to the Summary Technical 
Report of the Applied Mathematics Panel. 


CONFIDENTIAL^ 




Chapter 6 

SEABORNE FIRE CONTROL WITH RADAR 




S ECTION 7.6, charged with Navy fire control 
with radar, was formally organized very 
late in World War II (January 1944). Thus in 
view of the late start, the researches and devel- 
opments undertaken after formal section or- 
ganization necessarily were destined for frui- 
tion after the cessation of hostilities. In view 
of its field of activity the responsibilities for 
projects involving radar already underway 
were inherited by the section at the time of its 
formation. Inasmuch as the Chief of Section 
7.6 was a member of Division 7 from the date 
of NDRC reorganization, the four or five proj- 
ects begun before January 1944, which were 
his responsibility, might be said to carry this 
date, informally, back another year. Thus, 
despite the technical accomplishments which 
are to be recorded below it was the feeling of 
the section that its contribution towards win- 
ning the war was nil. 

In turning to the technical scene, one must 
return to the days of Section D-2 to summarize 
the radar developments sponsored by the divi- 
sion. 

6 1 DEVELOPMENT OF THE RADAR 
SCR-547 (PROJECT 14) 

During the early months of World War II, 
optical range and height finder equipment was 
a necessity and had no real competition in 
terms of equipment '‘in being” which met mili- 
tary requirements for supplying fire-control 
data. It was considered essential during the 
interim period of radar development to bring 
existing optical instruments to the highest pos- 
sible level of performance. Furthermore, ele- 
mentary consideration of prudence demanded 
that the situation be strongly hedged against 
two possibilities: (1) that unforeseen develop- 
ment, production, or training difficulties might 
seriously delay the application of radar as a 
means for obtaining fire-control data under the 
highly mobile conditions of field use; (2) that 
the enemy might develop effective countermeas- 

♦ 


ures. Hence Section D-2 vigorously pursued a 
program of optical range finder development. 
(See Section 2.6, and also Volume 2.) 

But Section D-2 did not rest with a program 
of mere “watchful waiting” with respect to 
radar. In January 1941, a request was made to 
Section D-1 for the development of radar 
equipment to provide range data for antiair- 
craft fire-control systems. In view of satura- 
tion of the facilities of the Radiation Labora- 
tory at the Massachusetts Institute of Techno- 
logy, Section D-1 recommended that Section D-2 
undertake to develop a range-only radar under 
separate contract with another laboratory. 

Accordingly, there was developed at the Bell 
Telephone Laboratories under contract NDCrc- 
156, a 10-cm transmitter pulsing at 400 cycles 
per second, and transmitting from a 54-in. 
parabolic antenna. A second 54-in. parabola re- 
ceives the reflections. The antennas and other 
radio equipment are mounted on a modified 
M2A4 sound-locator trailer with telescopes and 
controls for tracking in azimuth and elevation. 
Seats are provided for trackers and range op- 
erator (Figure 1). Because of the physical 
appearance of this device it was nicknamed 
“Mickey Mouse.” 

Range was measured by means of calibrated 
phase shifters which position the reflection 
with respect to a “step” in the timing trace on 
a cathode-ray screen. A full-range scale using 
a 4,000-c sweep displays all reflections from 
the 2,500-yard minimum to 41,600 yards. For 
precision measurements a 100,000-c sweep is 
used. This sweep voltage is obtained from a 
phase shifter which is geared mechanically 
(25:1 ratio) to the phase shifter operating at 
4,000 c so that the revolutions of the 100-kc 
phase shifter are counted with respect to the 
phase of the pulsing frequency. The readings of 
dials mounted on the two phase shifters are 
combined to read range, or the positions may 
be transmitted to coarse-fine selsyns. 

The experimental equipment was tested at 
Fort Monroe during the period from July 13, 


CONFIDENTIA 




62 



DEVELOPMENT OF THE CHRONOGRAPH T4 


63 


1941 to August 20, 1941. These tests included 
range measurements on fixed targets, surface 
vessels, casual aircraft, and target planes dy- 
ing missions. Details of these tests are sum- 


OEMsr-983 was negotiated with the Westing- 
house Electric and Manufacturing Co. for this 
purpose. 

This study showed that portable field chrono- 



Figure 1. SCR-547 antiaircraft range-finding equipment. 

This radar is located on a trailer and requires three operators. Men seated on the left and right of the apparatus 
use telescopes to keep the two parabolic antennas on the target. A third man is the radar operator. For obvious reasons 
this device was nicknamed “Mickey Mouse.” 


marized in the report of the Coast Artillery 
Board on Project 1213, August 23, 1941. In 
general, the results indicated that the probable 
error of range measurement with the optically 
tracked radio equipment was about % that of 
a stereoscopic height finder. There was, of 
course, the further advantage that the range 
data from the range finder are reasonably 
smooth so that a good range rate is available. 

This instrument was subsequently standard- 
ized by the Army as the radar SCR-547.^ 

62 DEVELOPMENT OF THE CHRONO- 
GRAPH T4 (PROJECTS 65 AND 83) 

In December 1942 a suggestion was advanced 
that Division 7 sponsor a study to determine 
the practicability of measuring the velocity of 
90-mm projectiles near the muzzle of the gun 
by means of the Doppler effect with continu- 
ous-wave radar. In January 1943 contract 


graphs could be made based on this principle.’ 
Under contract OEMsr-1405 (Project 83) 17 
engineered units were built for the Armed 
Services.^ 

The general scheme was as follows: Micro- 
wave energy is projected along the trajectory 
near the muzzle from a paraboloid antenna lo- 
cated near the gun. Some of this energy is 
intercepted and re-radiated by the projectile in 
flight. Because of the Doppler effect the fre- 
quency of the reflected energy differs from that 
of the energy radiated by the antenna by an 
amount proportional to the projectile velocity 
and the frequency of the radiated energy. Part 
of the reflected energy is picked up by a receiv- 
ing paraboloid located adjacent to the trans- 
mitter. This energy is conveyed to a crystal 
mixer where it is mixed with some of the 
directly transmitted energy to produce a volt- 
age of Doppler frequency which is a measure 
of projectile velocity. This voltage is amplified 


EbNFIDEiNTIAL| | 


64 


SEABORNE FIRE CONTROL WITH RADAR 




Figure 2. T-4 chronograph utilizing the Doppler effect. 

This radar chronograph transmits an r-f signal which is reflected by a moving projectile. The received frequency is 
heterodyned with the transmitter frequency in the unit to produce the Doppler frequency. The time duration of an arbi- 
trarily selected number, or train, of Doppler cycles is measured by special frequency and time-counting circuits. The data 
thus obtained is converted into velocity in feet per second by a simple calculation. A shows the r-f unit containing two 
paraboloid antennas to the left, the power supply unit in the center, and the counter unit to the right. B shows the r-f 
unit with cover removed showing the two paraboloid antennas. 




SEABORNE TORPEDO DIRECTORS 


65 


and then recorded by an electronic counter 
(Figures 2A and 2B). 

For the purpose of the investigation, radia- 
tion of 3-cm wavelength is used. For a projec- 
tile velocity of 2,850 ft per sec this gives a 
Doppler frequency of approximately 58,000 c. 
This frequency is amplified. A finite train of 
say 256 cycles actuates a counter. During this 
interval another electronic counter counts the 
number of cycles of a 200-kc crystal oscillator. 
The number of 200-kc counts during the one 
Doppler-frequency train is inversely propor- 
tional to the Doppler frequency. A small cor- 
rection for geometry is required to give the 
actual muzzle velocity. 

Preliminary tests indicated that two 15-in. 
parabolas, one transmitting 20 milliwatts of 
power, are adequate for ranges up to 600 feet 
on 90-mm shells. Oscillograms were obtained 
which showed that the limits of accuracy of the 
measurement of velocity are set only by the 
accuracy of measurement of the transmitter 
frequency and the Doppler frequency. With the 
counter method of recording, 14 per cent ac- 
curacy was obtained. 

The effects of gun flash and shock were 
investigated and satisfactory tests made at 
Aberdeen and Dahlgren. 

Under contract OEMsr-1404 with the Balti- 
more Plant of the Westinghouse Electric and 
Manufacturing Co., 17 experimental models of 
the chronograph T4 were constructed.^ These 
were delivered to the U. S. Army and Navy, to 
the British, and to the Civil Aeronautics Ad- 
ministration. 

One of the production models, prior to its 
acceptance, together with the original labora- 
tory model was taken by the Bureau of Ord- 
nance to the Pacific for measurements on the 
muzzle velocity of 16-in. guns aboard battle- 
ships. Successful and accurate measurements 
of the muzzle velocity were realized on about 
two-thirds of the rounds fired. Weaknesses in 
the production prototype were discovered and 
corrected in all the production units. The only 
serious difficulty encountered in these tests 
aboard ships was the triggering of the counting 
circuits by X-band radars. It was found 
necessary to turn off the ship’s X-band radars 
as well as the X-band radars of all other ships 


in the vicinity during the tests. The contractor 
recommended new development at a much 
shorter wavelength, where no radars are now 
contemplated. This recommendation was passed 
on to the Services. 


63 SEABORNE TORPEDO DIRECTORS 
(PROJECT 72) 

Following the development of the Mark 32 
airborne director (see Volume 3), the Navy re- 
quested Section 7.2 to adapt that development 
to the motor torpedo boat. The official request 
(NO-134) specified a director with a course 
and speed sight and also with blind-firing 
equipment. The coordination of this project 
with the Radiation Laboratory under Division 
14 disclosed the development of a simplified 
device, perhaps best described as a mechanized 
maneuvering board. It was agreed at a meet- 
ing of representatives of Divisions 7 and 14 
that it would be well to have two independent 
attacks on the problem. The Navy concurred 
and contract OEMsr-1208 was negotiated with 
the General Electric Co. 

Under the project a preliminary working 
model with a mechanical computer of the 
simplest possible type was constructed.^*'^ This 
computer permitted firing of the torpedo from 
a motor torpedo boat at any time the captain 
of the boat wished to bring the direction of his 
own course along a continuously calculated 
value. This director permitted avoiding action 
and firing on the run. Tests at Miami, Florida, 
indicated that the predictions were quite 
erratic. Several sources of error existed, such 
as the radar, the data-transmission system 
from the antenna to the PPI, the compass, the 
autosyn transmission of own-ship’s course, and 
the flexible shafts from the computer to the 
radar. No improvement to the system could be 
made until most of these errors were elimi- 
nated. 

A program was instituted to remedy the 
sources of error: first, by the use of K-band 
radar; and secondly, by redesign of those por- 
tions of the system which had shown up as 
sources of error in the Florida tests. Although 
work had begun on the assembly of the new 


/c ^TIDENTIAlI 


66 


SEABORNE FIRE CONTROL WITH RADAR 


system, the conclusion of World War II pre- 
vented completion of the project. In conform- 
ity with the termination policy as set up by 
OSRD, the Bureau of Ordnance [BuOrd] was 
given an opportunity to take over this project. 
BuOrd indicated that it wished to review its 
entire program in the light of changed circum- 
stances brought about by the end of war and 
that they deemed it desirable to reconsider the 
entire fire-control program for motor torpedo 
boats. The Navy will undertake, therefore, to 
set up a project at the Naval Research Labora- 
tory for further mathematical analysis of the 
problem. It was agreeable to the BuOrd that 
this project be terminated. The project was 
therefore terminated uncompleted on Septem- 
ber 30, 1945. 

BuOrd also asked NDRC in NO-197^ to de- 
velop a better torpedo director for destroyers 
than the Mark 27 which was then in use* and to 
make it blind firing. Examination of the prob- 
lem led to the conclusion that the director 
designed for motor torpedo boats was not ade- 
quate for a destroyer, and that more experi- 
ments would be needed. Theoretical studies 
were made to devise a director which would not 
be too cumbersome and large but would have 
the required accuracies. A satisfactory solution 
was found and design work was begun. With 
the termination of the contract, the work was 
continued under a direct BuOrd contract with 
the General Electric Co. 

^ REDESIGN OF GUN DIRECTOR 
MARK 49 

When director Mark 49 was placed in service 
unfortunate failures of some component parts 
resulted in the cancellation of the contracts 
for its production. Upon Navy invitation the 
section made a study which indicated that if 
certain changes were made the director would 
probably be an acceptable piece of naval equip- 
ment. 

One of the outstanding difficulties with the 
Mark 49 was the failure of the clutch-type 
power drive after a relatively small number of 
hours of operation. The take-off of the rate- 
measuring gyro was crude and entirely un- 
satisfactory in operation. There were also a 


number of other bad features such as restricted 
angle of vision of the operator and maximum 
lead angle, inadequate radar reflector, and poor 
radar indication. 

Under the original request it was understood 
that NDRC would attempt to make whatever 
modifications were necessary in the gun direc- 
tor Mark 49 system to increase its usefulness 
as a blind-firing unit. As a result of subsequent 
discussions the program was narrowed in 
scope, so as to include only modifications in 
the power drive and gyro element. 

Under contract OEMsr-1235 the Servomecha- 
nisms Laboratory at the Massachusetts Institute 
of Technology installed amplidyne power drives 
in one unit. The contactor take-off on the gyro 
was replaced by an induction take-off.® The 
resulting system has a maximum angular rate 
in train of 35 degrees per sec and maximum 
acceleration in the neighborhood of 100 degrees 
per sec per sec. A modified gun director Mark 
49 was delivered to the Naval Test Center at 
Dam Neck, Virginia. 

In spite of the very satisfactory tracking 
and drive characteristics obtained in the modi- 
fied director, these modifications were not put 
into Service use largely because the number of 
Mark 49 directors for which such modification 
procedure was practicable was too small to 
justify the cost. 

6 5 GUNFIRE-CONTROL SYSTEM 
MARK 56 (PROJECTS 71, 79, 85) 

At the beginning of 1944 the record of the 
Navy fire-control development in antiair- 
craft was as follows: (1) in the field of heavy 
antiaircraft guns no new fire-control equip- 
ment had been introduced into the Navy since 
the beginning of World War II ; (2) in the field 
of automatic weapons, the director Mark 51 
had been introduced and some improvement 
was on the way; (3) the heavy antiaircraft gun 
directors employed no radar with a beam less 
than about 16 degrees in width; and (4) no 
microwave equipment was getting into the 
fleet for one reason or another except as an 
antiaircraft set for use with directors Mark 33 
on certain cruisers. It was abundantly clear 
to Division 7 members and their employees 



GUNFIRE-CONTROL SYSTEM MARK 56 


67 


that the Navy was in need of a modern radar- 
equipped director capable of employing radar 
data, of getting quick solutions, and of per- 
forming the calculations with an accuracy ade- 


including radar, by patching up a system al- 
ready in existence in the Navy, convinced the 
members of the Radiation Laboratory and 
members of Division 7 of NDRC that a fully 


FUSE 


DEAD TIME AVERAGE BULLET 



quate over all ranges of speeds of modern 
planes. It was under these circumstances that 
the NDRC project known as Gunfire-Control 
System Mark 56 was started. (See Figures 3 
and 4.) 

It is desirable to state briefly the history of 
the gunfire-control system project up to the 
time of the formation of Section 7.6 of NDRC.^ 
The failure of NDRC to provide an adequate 
antiaircraft fire-control system for the Navy- 


integrated design of a complete fire-control 
system was required. Great fear was expressed 
in many quarters that the enemy might de- 
velop aerial weapons with which neither the 
director system Mark 37 nor its radars could 
cope. Section T of OSRD that had just success- 
fully completed the proximity fuze develop- 
ment, apparently reached the same conclusions 
at about the same time. Therefore, to assist the 
Navy, a deliberate program was established by 



68 


SEABORNE FIRE CONTROL WITH RADAR 


NDRC to provide a completely integrated radar 
fire-control system. This project was to be 
carried through to complete engineering design, 
the building of manufacturing prototypes, the 
furnishing of complete manufacturing draw- 
ings, and the setting up of a manufacturing 
source and all necessary vendors. This ap- 
proach was necessitated by the past experiences 
in which laboratory prototypes had been found 
wanting not operationally, but because of the 
long time element between laboratory research 
and their subsequent engineering for produc- 
tion. The plan required the following steps : 

1. Complete support from the heads of Divi- 
sions 7 and 14 of NDRC. 

2. Complete cooperation of the Director of 
Radiation Laboratory. 

3. The sympathetic understanding by the 
heads of NDRC and OSRD. 

4. The establishment of adequate contract- 
ing with a company able to assist in the engi- 
neering and also able to manufacture. 

5. The sponsorship and cooperation of the 
Bureau of Ordnance. 

The Bureau of Ordnance indicated its sup- 
port by requesting the establishment of the 
project NO-166 in a letter, dated May 18, 1943. 

The General Electric Co. was brought into 
the picture by an NDRC contract, first for the 
development of a suitable gyro unit (OEMsr- 
1181), and later for supplying engineering, 
parts, and the construction of two complete 
director systems (OEMsr-1299). 

Under contract OEMsr-1181 the General 
Electric Co. cooperated in the development of a 
line-of-sight gyro and a vertical gyro which 
supplied both stabilization and regular rate 
data for the gun director Mark 56.® 

' The gun director Mark 56 (Figure 3) is 
for use primarily with 5-inch dual-purpose guns 
with emphasis on low-flying torpedo plane 
attacks. It is a fully automatic blind-firing unit 
with emphasis on ruggedness, ease of main- 
tenance, and speed of response. 

As noted above, this development was 
sponsored cooperatively by Divisions 7 and 14 
of NDRC. Development work was done both 
at the Radiation Laboratory under contract 
OEMsr-262 and by the General Electric Co. 
under contract OEMsr-1299. Component work 


was done at both laboratories and a first labo- 
ratory unit designated gun director Mark 56 
Model 0 was set up at the Radiation Labora- 
tory. Two additional laboratory models were 



Figure 4. Antenna of the Mark 56 radar. The 
X-band radar is fully automatic in all coordi- 
nates. The stabilization of the disk is apparent 
in the figure by the horizontal position of the 
antenna feed pointing to the horizon in contrast 
to the tilt of the deck. 

built, followed by two prototypes from manu- 
facturing drawings. Under the terms of the 
contract the General Electric Co. built these 
two complete prototype directors based on the 
laboratory models. 

The gun director Mark 56 employs a two- 
axis mount.^ Rate stabilization and tracking is 
performed through a gyro unit assembly. This 
unit fixes rotation around a line of sight giving, 
so far as prediction is concerned, a three-axis 
system. Bullet velocity is computed explicitly 
and is used in calculating angular deflections. 
The angular deflections are added to the present 
position and the sums are converted to future 
gun coordinates by a mechanical system of 
bevel gears. 

An X-band radar fully automatic in all coor- 
dinates (Figure 4) is used. Spiral scan is in- 
corporated. Emphasis was placed on antijam- 
ming features and on built-in test equipment. 

The termination policy of OSRD did not 
permit the completion of this program. It was 
therefore transferred to BuOrd, which nego- 


GUNFIRE-CONTROL SYSTEM MARK 56 


69 


tiated a new contract with the General Electric 
Co. for the completion of the units 4 and 5, for 
which it had prime responsibility. The contract 
of BuOrd became effective on October 28, 1945, 
and the NDRC project terminated on that date. 
The new contract is of the task type, under 
which provisions are made for coordinating the 
work at the Librascope Corp. (ballistic com- 
puter), and for providing assistance to the 
Navy in maintaining the two units for which 
the Radiation Laboratory had prime respon- 
sibility (OEMsr-262). The two units which the 
Radiation Laboratory delivered to the Navy 
were physically complete, but not fully “de- 
bugged” or tested. 

Prior to the end of World War II, the Navy 
had placed a letter of intent with General Elec- 
tric for the construction of a substantial quan- 
tity of the Gunfire-Control System Mark 56. 
The cessation of hostilities did not bring about 
a cancellation. 

Contract OEMsr-1044 with the Librascope 
Corp. was transferred from Division 14 to 
Division 7 when it became evident that the 
bulk of this contract covered a part of the com- 
puter for the gun director Mark 56. The por- 
tion of the computer which Librascope Corp. 
manufactured was called the ballistic com- 
puter.^® It is a mechanical computer using 
levers for addition, multiplication, division, and 
ballistic functions. Its inputs are driven by 
remote transmission from the gun director 
Mark 56 and its associated radar Mark 35. It 
uses these observed quantities to compute cer- 
tain functions such as u, the average bullet 
velocity to future position ; g, the time of flight 


were there no cross components of the target 
velocity ; and various other functions. The out- 
puts of the computer are voltages. The voltages 
are used in electrical networks solving for gun 
deflections, unit parallax, and wind corrections. 
Hand inputs to the computer provide for true 
wind, dead time, initial velocity, density, and 
spot correction. The computer is not complete 
by itself. On the other hand, because of the in- 
tegration of the gun director Mark 56 system, 
it contains parts of the radar and data-trans- 
mission systems. 

A breadboard model of the computer was 
delivered and tested for Class A errors as well 
as for effects of vibration. Using these tests as 
checks, detailed specifications were set up with 
Librascope Corp. to guarantee the success of 
the computer from all Service standpoints. 

Five prototype models were required by this 
contract. 

In order to provide proper design and in- 
tegration of this unit with the rest of the Mark 
56 system, personnel from the General Electric 
Co. under contract OEMsr-1299 and the Radia- 
tion Laboratory under contract OEMsr-262 were 
made available to the Librascope Corp. 

This project was terminated by NDRC as of 
October 31, 1945. BuOrd placed a new contract, 
effective the same date, for the continuation of 
the subject work. In order to provide for engi- 
neering coordination with the General Electric 
Co., which is to be responsible for the overall 
system, the new contract instructed the Libra- 
scope Corp. to deal directly with the General 
Electric Co. on engineering matters and with 
BuOrd on legal and fiscal matters. 


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PART II 


DATA SMOOTHING AND PREDICTION IN FIRE-CONTROL SYSTEMS 


By R. B. Blackman, H. W. Bode, and 
C. E. Shannon ^ 


T he problem of data smoothing in fire con- 
trol arises because observations of target 
positions are never completely accurate. If the 
target is located by radar, for example, we may 
expect errors in range running from perhaps 
10 to 50 yards in typical cases. Angular errors 
may vary from perhaps one to several mils, 
corresponding at representative ranges, to 
yardage errors about equal to those mentioned 
for range. Similar figures might be cited for 
the errors involved in optical tracking by vari- 
ous devices. Evidently these errors in observa- 
tion will generate corresponding errors in the 
final aiming orders delivered by the fire-control 
system. 

A data-smoothing device is a means for mini- 
mizing the consequences of observational er- 
rors by, in effect, averaging the results of ob- 
servations taken over a period of time. The 
simplest example of data smoothing is fur- 
nished by artillery fire at a fixed land target. 
Here the principal parameter is the range to 
the target. While individual determinations of 
the range may be somewhat in error, a reliable 
estimate can ordinarily be obtained by taking 
the simple average of a number of such ob- 
servations. This example, however, is scarcely 
a representative one for problems in data 
smoothing generally. The errors involved are 
small and the averaging process is an elemen- 
tary one. Moreover, the data-smoothing proc- 
ess is not of very decisive importance in any 
case, since any errors which may exist in the 
estimated range can normally be wiped out 
merely by observing the results of a few trial 
shots. 

More representative problems in data 
smoothing arise when we deal with a moving 
target. In this case errors in observational 
data may be much more serious, since they 
determine not only the present position of the 
target but also the rates used in calculating 
how much the target will move during the time 
it takes the projectile to reach it. An illustra- 
tion is furnished by antiaircraft fire against 


» Bell Telephone Laboratories. 


distant airplanes. Suppose, for example, that 
in observing the target’s position we make two 
errors of opposite sign and a second apart, of 
25 yards each. Then the apparent motion of 
the airplane is in error by 50 yards per second. 
Since the time of flight of an antiaircraft shell 
in reaching its target may be as high as 30 
seconds or more, such an error might produce 
a miss of the order of 1 mile. It is clear that 
in any comparable situation the effect of ob- 
servational errors in determining the target 
rate will be much greater than the position er- 
ror alone would suggest, and the function of 
the data-smoothing network in averaging the 
data so that even moderately reliable rates can 
be obtained as a basis for prediction becomes 
a critically important one. 

Aside from magnifying the consequences of 
small errors in target position, the motion of 
the target complicates the data-smoothing 
problem in two other respects. The first is the 
fact that it gives us only a brief time in which 
to obtain suitable firing orders. The total en- 
gagement is likely to last for only a brief time, 
and in any case it is necessary to make use of 
the data before the target has time to do some- 
thing different. Thus the averaging process 
cannot take too long. The second complication 
results from the fact that the true target posi- 
tion is an unknown function of time rather 
than a mere constant. Thus many more possi- 
bilities are open than would be the case with 
fixed targets, and the problem of averaging 
to remove the effects of small errors is cor- 
respondingly more elusive. 

The intimate relation between data smooth- 
ing and target mobility explains why the data- 
smoothing problem is relatively new in war- 
fare. The problem emerged as a serious one 
only recently, with the introduction of new and 
highly mobile military devices. The airplane is, 
of course, the archetype of such mobile instru- 
ments, and we have already mentioned the 
data-smoothing problem as it appears in anti- 
aircraft fire. Since the relative velocity of air- 
plane and ground is the same whether we sta- 
tion ourselves on one or the other, however, the 




;0.\TIDE.MlAr. 


D 


71 


72 


DATA SMOOTHING AND PREDICTION IN FIRE-CONTROL 


mobility of the airplane produces essentially 
the same sort of problem in the design of bomb- 
sights also. Another field exists in plane-to- 
plane gunnery. Although they are somewhat 
slower, the mobility of such vehicles as tanks 
and torpedo boats is still considerable enough 
to create a serious problem here also. Future 
examples may be centered largely on robot 
missiles. It is interesting to notice that a 
guided missile may present a problem in data 
smoothing either because it belongs to the 
enemy, and is therefore something to shoot at, 
or because it belongs to us, and requires 
smoothing to correct errors in the data which 
it uses for guidance. The tendency to higher 
and higher speeds in all these devices must 
evidently mean that fire control generally, and 
data smoothing as one aspect of fire control, 
must become more and more important, unless 
war making can be ended. 

Very mobile instruments of war, such as 
the airplane, began to make their appearance 
in World War I, but there was insufficient time 
during that war to make much progress with 
the fire-control problems which such instru- 
mentalities imply. In the interval between 
World War I and World War II, however, a 
considerable number of fire-control devices, 
such as bombsights and antiaircraft compu- 
ters, were developed. The principal attention 
in the design of these devices, however, was 
on the kinematical aspects of the situation. 
Although a number of them included fairly 
successful methods of minimizing the effects of 
observational errors,*" it seems fair to say that 
in the interval between the two wars there 
was no general appreciation of the existence of 
the data-smoothing problem as such. 

It follows that the theory of data smoothing 
advanced in this monograph is the result prin- 
cipally of experience gained in World War II. 
More specifically, it is the product of the ex- 


^ Most of these solutions depended upon the use of 
special types of tracking systems. Examples are found 
in the use of regenerative tracking in bombsights and 
antiaircraft computers or in the determination of rates 
from a processing gyroscope or an aided laying mech- 
anism in an antiaircraft tracking head. So far as their 
effect on the data-smoothing characteristics of the 
overall circuit is concerned, these devices are equiva- 
lent to simple types of smoothing networks inserted 
directly in the prediction system. This is discussed in 
more detail under the heading “Exponential Smooth- 
ing,” Section 10.1. 


perience of the authors with a series of proj- 
ects, largely sponsored by Division 7 of NDRC, 
concerned with the design of electrical antiair- 
craft directors. In addition, it draws largely 
on the results of a number of other investiga- 
tions, also NDRC sponsored. The possible key 
importance of data smoothing in the design of 
fire-control systems was recognized by Division 
7 early in the course of its activities and the 
emphasis placed upon it in a number of proj- 
ects led to the accumulation of a much larger 
body of results than might otherwise have been 
obtained. 

Data smoothing is developed here in terms 
of concepts familiar in communication engi- 
neering. This is a natural approach since data 
smoothing is evidently a special case of the 
transmission, manipulation, and utilization of 
intelligence. The other principal, and perhaps 
still more fundamental, approach to data 
smoothing is to regard it as a problem in sta- 
tistics. This is the line followed in the classic 
work^ by Norbert Wiener.'" For reasons which 
are brought out later, Wiener’s theory is not 
used in the present monograph as a basis for 
the actual design of data-smoothing networks. 
Because of its fundamental interest, however, 
a sketch of Wiener’s theory is included. The 
authors’ apologies are due for any mutilation 
to the theory caused by the attempt to simplify 
it and compress it into a brief space. 

The present monograph falls roughly into 
two dissimilar halves. The first half, consist- 
ing of the first three or four chapters, includes 
a discussion of the general theoretical founda- 
tions of the data-smoothing problem, the best 
established ways of approaching the prob- 
lem, the assumptions they involve, and the 
authors’ judgment concerning the assumptions 
which best fit the tactical facts. In this part 
may also be included the last chapter, which 
contains a fragmentary discussion of alterna- 
tive data-smoothing possibilities lying outside 
the main theoretical framework of the mono- 
graph. 

The rest of the monograph is concerned with 
the technique of designing specific data-smooth- 
ing structures. A fairly elaborate and detailed 
treatment is given here, in the belief that the 


Wiener is also responsible for providing tools which 
permit the gap between the statistical and communica- 
tion points of view to be bridged. 


^ CO.M 



73 


DATA SMOOTHING AND PREDICTION IN FIRE-CONTROL 


problem of actually realizing a suitable data- 
smoothing device is, in some ways at least, 
as difficult as that of deciding what the general 
properties of such a device should be. The 
technique, as given, draws heavily upon the 
highly developed resources of electric network 
theory. For this reason the discussion is 
couched entirely in electrical language, al- 
though the authors realize, of course, that 
equivalent nonelectrical solutions may exist. 
For the benefit of readers who may not be 
familiar with network theory, the monograph 
includes an appendix summarizing the prin- 
ciples most needed in the main text. 

Two further remarks may be helpful in un- 
derstanding the monograph. The first concerns 
the relation between data smoothing and the 
overall problem of prediction in a fire-control 
circuit. These two are coupled together in the 
title of the monograph, and it is clear that the 
connection between them must be very close, 
since, as we saw earlier, small irregularities in 
input data are likely to be serious only as they 
affect the extrapolation used to determine the 
future position of a moving target. In the 
statistical approach, in fact, data smoothing 
and prediction are treated as a single problem 
and a single device performs both operations. 

In the attack which is treated at greatest 
length in the monograph a certain distinction 
between data smoothing and prediction can be 
made. To simplify the exposition as much as 
possible, the explicit discussion in the mono- 
graph is directed principally at data smooth- 
ing. This, however, is not intended to suggest 
that there is any real cleavage between the 
two problems or that the analysis as developed 
in the monograph does not also bear, by impli- 
cation, upon the prediction problem. Any the- 
ory of data smoothing must rest ultimately 
upon some hypothesis concerning the path of 
the target, and the exact statement of the as- 
sumptions to be made is in many ways the most 
important as well as the most difficult part of 
the problem. The same assumptions, however, 
are also involved in the extrapolation to the 
future position of the target. It is thus impos- 
sible to solve the data-smoothing problem with- 
out also implying what the general nature of 
the prediction process will be. For example, 
the formulation given in Chapter 9 amounts to 


the assumption that the target path is specified 
by a set of geometrical parameters correspond- 
ing to components of velocity, acceleration, etc. 
The data-smoothing process centers about the 
problem of obtaining reliable values for these 
parameters. To obtain a complete prediction 
thereafter, it is merely necessary to multiply 
the parameter values thus obtained by suitable 
functions of time of flight and add the results 
to the present position of the target. 

The other general remark concerns the tacti- 
cal criteria used in evaluating the performance 
of a data-smoothing system. This turns out to 
be one of the most important aspects of the 
whole field. It is assumed here that the tactical 
situation is similar to that of antiaircraft fire 
against high-altitude bombers in World War 
II. The defense can be regarded as successful if 
only a fairly small fraction of the targets en- 
gaged are destroyed. On the other hand, the 
lethal radius of the antiaircraft shell is so small 
that it is also quite difficult to score a kill. 
Under these circumstances we are interested 
only in increasing the number of very well 
aimed shots. 

When we combine these assumptions with 
the path assumptions described in Chapter 9 
we are led to the data-smoothing solution for- 
mulated here, in preference to the solution ob- 
tained with the statistical approach. On the 
other hand, we might equally well envisage a 
situation in which the target contained an 
atomic bomb or some other very destructive 
agent, so that it becomes very important to 
intercept it, while the lethal radius of the anti- 
aircraft missile is correspondingly increased, 
so that great accuracy is not needed for a kill. 
In this situation our interest would be focused 
on the problem of minimizing the probability 
of making large misses, and the solution fur- 
nished by the statistical approach would be ap- 
proximately the best obtainable.^ 


In fairness to the statistical solution it should be 
pointed out that it is also the best obtainable, without 
regard to the lethal radius of the shell, if we replace 
the path assumptions made in Chapter 9 by a “random 
phase” assumption. The path assumptions in Chapter 
9 are almost at the opposite pole from a random phase 
assumption, and represent a deliberate overstatement, 
made in order to illustrate the theoretical situation as 
clearly as possible. 




. .r' 




Chapter 7 


GENERAL FORMULATION OF THE DATA-SMOOTHING PROBLEM 


O NE OF THE PRINCIPAL difficulties in any 
treatment of data smoothing is that of 
stating exactly what the problem is and what 
criteria should be applied in judging when we 
have a satisfactory solution. It is consequently 
necessary to embark upon a rather extensive 
general discussion of the data-smoothing prob- 
lem before it is possible to consider specific 
methods of designing data-smoothing struc- 
tures. This preliminary survey will occupy 
Chapters 7, 8, and 9. As a first step this chap- 
ter will describe two of the general ways in 
which the data-smoothing problem can be ap- 
proached mathematically. The formulation of 
the problem which is finally reached in Chap- 
ter 9 is not the one which is most obviously 
suggested by these approaches. This, however, 
does not lessen their value in characterizing 
the problem broadly. 

7 1 A PHYSICAL ILLUSTRATION 

In an actual fire-control system the data- 
smoothing problem is usually made fairly spe- 
cific because of the particular geometry 
adopted in the computer. It may be helpful 
to have some particular case in mind as a 
touchstone in interpreting the general discus- 
sion. For this purpose the most appropriate 
example is furnished by long range land-based 
antiaircraft fire, since most of the analysis 
described in this monograph was developed 
originally for its application to this problem. 
It is usually assumed in the antiaircraft prob- 
lem that the target flies in a straight line at 
constant speed, and in one case at least the 
computer operates by converting the input data 
into Cartesian coordinates of target position 
and differentiating these to find the rates of 
travel in the several Cartesian directions. 
These rates form the basis of the extrapolation. 

The process is illustrated in Figure 1. The 
input coordinates are transformed into elec- 
trical voltages proportional to Xp, yp, and Zpy 
the Cartesian coordinates of present position, 



in the coordinate converter at the left of the 
diagram. The extrapolation for x is shown 
explicitly. It consists essentially in differen- 
tiating to find the x component of target 
velocity, multiplying the derivative by the time 
of flight tf and adding the result to Xp to find 



Figure 1. Data-smoothing networks in linear pre- 
diction circuit. 


Xp, the predicted future value of x. A similar 
procedure fixes Vf and Zp. After the addition 
of certain ballistic corrections, these three co- 
ordinates of future position are transformed 
into gun aiming orders in the coordinate con- 
verter shown at the right of the drawing. This 
last unit also provides the time of flight re- 
quired as a multiplier in the extrapolation. 

The small irregularities in the input data 
caused by tracking errors are greatly magni- 
fied by the process of differentiation. It is thus 
necessary to smooth the rates considerably if 
a reliable extrapolation is to be secured. The 
data-smoothing network for the x coordinate is 
represented by in Figure 1. Since the Car- 
tesian velocity components are theoretically 
constants if the assumption of a straight line 
course at constant speed is correct, a data- 
smoothing network in this computer must be 
essentially an averaging device which gives 
an appropriately weighted average of the fluc- 
tuating instantaneous rate values fed to it. The 
problem of “smoothing a constant*’ is given 
special attention in Chapter 10. Aside from the 
particular circuit of Figure 1, we may, of 
course, be required to smooth a constant when- 
ever the prediction is based upon an assumed 
geometrical course involving one or more pa- 
rameters which are isolated in the circuit. 



75 


76 


FORMULATION OF THE DATA-SMOOTHING PROBLEM 


In addition to smoothing the rates we can, 
if we like, attempt to smooth the irregularities 
in present position also. A network to accom- 
plish this purpose is indicated by the broken 
line structure in Figure 1. Of course, in 
dealing with the present position we are no 
longer smoothing a constant, but suitable struc- 
tures can be obtained by methods described 
later. However, the effect of tracking errors in 
the present position circuit is so much less than 
it is in the rate circuit that can generally 
be omitted. 

Geometrical assumptions of the sort implied 
in Figure 1 are helpful in visualizing the prob- 
lem, and they are of course of critical impor- 
tance in determining what the final data- 
smoothing device will be. It is important not 
to make explicit assumptions of this kind too 
early in the formal analysis, however, since 
the meaning of such assumptions is one of the 
aspects of the general problem which must be 
investigated. For example, it is apparent that 
no airplane in fact flies exactly a straight line, 
nor flies a straight line for an indefinite period. 
In detail, the solution of the data-smoothing 
problem depends very largely on how we treat 
these departures from the idealized straight 
line path. For the present, consequently, it will 
be assumed that the input data are presented 
to the data-smoothing and predicting devices 
in terms of some generalized coordinates, the 
nature of which we will not inquire into too 
closely. A given coordinate might, for example, 
be a velocity, a radius of curvature, an angle of 
dive or climb, or any other quantity which 
would be directly useful in making a predic- 
tion, or it might be a simple position coordi- 
nate such as an azimuth or an altitude. 

The data-smoothing and predicting opera- 
tion itself is assumed to be performed by linear 
invariable devices. Aside from the fact that 
this assumption is, of course, a tremendously 
simplifying one, it also fits the data-smoothing 
problem very nicely, as the problem is formu- 
lated in this chapter. With other formulations, 
however, it appears that somewhat better re- 
sults may be obtainable from variable devices 
or devices including more or less radical 
amounts of nonlinearity. These possibilities are 
discussed briefly in Chapter 14. 


7 2 data SMOOTHING AND 

PREDICTION 

Figure 1 illustrates a distinction between 
two possible methods of looking at the data- 
smoothing problem which it is advisable to 
establish for future purposes. In describing the 
X system in Figure 1 we laid emphasis on the 
particular networks and N^. It is clear, how- 
ever, that the complete x circuit with input Xp 
and output Xp is a network having overall 
transmission properties which can be studied. 
Since tf will normally vary with time, the net- 
work is not, strictly speaking, an invariable 
one, but the changes of tf are ordinarily too 
slow to make this an essential consideration. 

When it is necessary to make a distinction 
between these points of view, a network such 
as Ai, which is merely an element in the pre- 
diction process, will be called a data-smoothing 
structure. An overall circuit, providing data 
smoothing and prediction in one step, will be 
called a data-smoothing and prediction net- 
work, or simply a prediction network. Al- 
though these points of view have been illus- 
trated for rectangular coordinates, they obvi- 
ously apply also in many other situations. For 
example, we might go so far as to apply the 
overall point of view to a complete circuit from 
input azimuth, say, to output azimuth. 

Both points of view are taken from time to 
time in the monograph. When possible, how- 
ever, principal attention has been given to the 
limited data-smoothing problem. This tends to 
simplify the discussion, since the limited prob- 
lem is evidently more concrete than the overall 
prediction problem. Moreover, it permits us to 
deal lightly with such questions as the particu- 
lar choice of coordinates in which the smooth- 
ing operations are conducted, since it assumes 
that the general kinematical framework of pre- 
diction has already been decided upon. On the 
other hand, the overall point of view is more 
effective in certain situations, and it is the only 
natural one in the statistical treatment de- 
scribed in the next section. 

7 3 DATA SMOOTHING AS A PROBLEM 
IN TIME SERIES 

The most direct and perhaps the most gen- 
eral approach to data smoothing consists in re- 


CONFIDENTTAL 


THE AUTOCORRELATION 


77 


garding it as a problem in time series. This 
is the approach used by Wiener in his well- 
known work.^ It essentially classifies data 
smoothing and prediction as a branch of statis- 
tics. The input data, in other words, are 
thought of as constituting a series in time 
similar to weather records, stock market prices, 
production statistics, and the like. The well- 
developed tools of statistics for the interpreta- 
tion and extrapolation of such series are thus 
made available for the data-smoothing and 
prediction problem. 

To formulate the problem in these terms, 
let fit) represent the true value of one of the 
coordinates of the target and let git) repre- 
sent the observational error. Then fit) and 
git) are both time series in the sense just 
defined. The set of all such functions corre- 
sponding to the various possible target courses 
and tracking errors form an ensemble of time 
series or a statistical population. One can im- 
agine that a large number of particular func- 
tions fit) and git) have been recorded, each 
with a frequency proportional to its actual 
frequency of occurrence. Wiener assumes that 
they are stationary, that is, that the statistical 
properties of the ensemble are independent of 
the origin of time. This, of course, implies that 
both functions exist from ooto^=+°°- 

We will sometimes find it more convenient to 
make the assumption that the two functions 
vanish after some fixed, but sufficiently remote, 
points on the positive and negative real t axis.* 

The input signal to the computer is of course 
fit) -f git). If we assume that the coordinate 
in question represents a position, the quantity 
we wish to obtain is fit + tf) , where tf repre- 
sents the prediction time. If the coordinate is 
a rate, we are interested in an average value 
of fit) over the prediction interval. This com- 
plicates the mathematics somewhat, but does 
not essentially affect the situation. 


‘ This is done for technical mathematical reasons. We 
shall later have occasion to consider the Fourier trans- 
forms of f(t) and g{t), and, to have well-defined trans- 
forms, the integrals of the squares of the two func- 
tions, from t=— CO to t=+co^ should be finite. This 
would not happen under the “stationary” assumption. 
Wiener avoids the difficulty by introducing what he 
calls a generalized harmonic analysis, but this method 
is far too complicated to be treated in a brief sketch 
like the present. 


We shall not, of course, be able to predict 
fit + tf) perfectly accurately. Let the pre- 
dicted value be represented by /*(f + f/). In 
virtue of our assumption that the data- 
smoothing and prediction circuit is to be a 
linear invariable network, the relation between 
f^it-^tf) and the total input signal fit) 
-^git) can be written as 

Fit -h </) = / [/w + gi<T)]dK{a) (1) 

-00 

where dKia) represents the effect of the data- 
smoothing and prediction circuit. Comparison 
to equations (2) and (5) of Appendix A shows 
that K is, in fact, the indicial admittance of 
this circuit. The particular problem to be 
solved is of course that of finding a shape for 
the function Kia) which will make Fit + tf) 
the best possible estimate of /(f + tf) . 

The fact that the upper limit of integration 
in equation (1) is taken as o- = 0 is particu- 
larly to be noted. It corresponds to the fact that 
in making a prediction we are entitled to use 
only the input data which has accumulated up 
to the prediction instant. This restriction will 
be conspicuous in the next chapter, where the 
time-series analysis is completed. 


THE AUTOCORRELATION 


The principal statistical tool used in study- 
ing equation (1) is the so-called autocorrela- 
tion. Under the “stationary"' assumption the 
autocorrelation for fit) i^ defined by 

rT 

<t>i{T) = lim i / /(/ + T)f{t)dt . (2) 

T-^ca J -T 


We can obtain a normalized autocorrelation, 
which is more convenient for some purposes, 
by dividing by <piiO). This gives 


<f>iN (r) 


4>i(t) 

0i(O) 



fit + r)mdt 


\fit)\Ut 


.(3) 


If we assume that fit) in fact vanishes for 
sufficiently large positive or negative values of 
t, the limit sign can be disregarded and (ji^Nir) 
becomes simply 


^ONFIDENTIALJ 


78 


FORMULATION OF THE DATA-SMOOTHING PROBLEM 


4>iN(r) 



+ r) fm 


(4) 


where Wi 



[f and represents the total 


“energy’’ in the time series /(<). 


Precisely similar expressions can be set up 
for the autocorrelation </) 2 (t) or <^ 2 a (t) of the 
observational error function g{t). In a gen- 
eral case we might also have to worry about 
a possible cross correlation between f{t) and 
g{t). This would be represented by a cross- 
correlation function <^i 2 (t)j obtained by inte- 
grating the product f (t t) g (t) . In practical 
fire control, however, it can be assumed that 
the correlation between target course and 
tracking errors is small enough to be neglected. 

As a simple example of the calculation of 
an autocorrelation we may assume that f(t)^ 
sin U. Then 


1 r'^ 

bi (r) = hm ^ / sin + 

T—^ 00 ^ ^ _'p 


= lira 9 ^ 

T-^co ^ 1 


■) sin oit • dt 


- cos (2cot -j- 03T)]d 


since the term cos (2o}t + wt) will contribute 
nothing in the limit. 

The maximum value of (piir) in (5) is found 
at T = 0. This is to be expected, since ob- 
viously the correlation between identical val- 
ues of the function is the best possible. What 
is exceptional about* the present result is the 
fact that is not small for all large r’s. 

This is fundamentally a consequence of the 
fact that we chose an analytic expression for 
f{t)f so that the relation between two values 
of the function is completely determinate, no 
matter how great the difference between their 
arguments. In a more representative time 
series, involving a certain amount of statisti- 
cal uncertainty, we would expect to ap- 

proach zero as r increases, refiecting the in- 
creasing importance of statistical dispersion as 
the time interval becomes greater. 

The significance of the autocorrelation func- 
tion for data smoothing and prediction is ob- 
vious without much study. Thus, suppose for 


simplicity that the observational error g(t) 
is zero. Then the autocorrelation is the 

only one involved. It is a measure of the ex- 
tent to which the true target path “hangs to- 
gether” and is thus predictable. For example, 
in weather forecasting it is a well-known prin- 
ciple that in the absence of any other infor- 
mation it is a reasonably good bet that tomor- 
row’s weather will be like today’s but that the 
reliability of such a prediction diminishes rap- 
idly if we attempt to go beyond two or three 
days. This would correspond to an autocorrela- 
tion function which is fairly large in the neigh- 
borhood of T = 0, but diminishes rapidly to zero 
thereafter. 

In a similar way the autocorrelation of the 
observational error g{t) represents the extent 
to which this error hangs together. In this 
case, however, a high correlation is exactly 
what we do not want. Thus, if </> 2 (t) vanishes 
rapidly as r increases from zero, closely neigh- 
boring values of g are quite uncorrelated, and 
we need only average the input data over a 
short interval in the immediate past in order 
to have most of the observational errors aver- 
aged out. If <^ 2 (t) is substantial for a much 
longer range, on the other hand, a much longer 
averaging period is necessary, with corre- 
spondingly greater uncertainties in the value 
obtained for f{t). 

THE LEAST SQUARES ASSUMPTION 

The autocorrelation function does not in it- 
self suffice to determine a time series com- 
pletely. For example, it is easily seen that the 
functions sin t + sin 2t and sin t + cos 2t have 
the same autocorrelation in spite of the fact 
that they represent waves of quite different 
shape. The autocorrelation function, however, 
has a peculiar importance in the fact that 
under many circumstances it is the only piece 
of information about the time series which we 
need to know. 

The significance of the autocorrelation be- 
comes apparent as soon as we investigate the 
error in prediction. In many mathematical sit- 
uations involving linear systems it is conven- 
ient to deal with the square of the error rather 
than with the error itself, since a first varia- 
tion in the error squared expression gives a 


CONI IDEMTAL 


DATA SMOOTHING AS A FILTER PROBLEM 


79 


linear relationship in the quantities of direct 
interest. We will deal with the square of the 
error here. If E represents the instantaneous 
error, /*(t + t/) — f{t + t;), the mean square 
error over a long period of time is evidently 

£2= lim ~ r [l*{l + t/)-f(t + tf)Ydt 

,r^«2TXr 

= lim — f [f {I tt)Y di 

T^oo 2T' J_j, 

1 

T / / ( ^ + ^/ ) /* ( ^ + ^/ ) ^^ 

T — >00 1 j' 

+ lim L r [f*{t + t,)Ydt. (6) 

T — >00 2T J _ rp 

The first integral in equation (6) can be 
evaluated immediately. From (2) it is 
To evaluate the second integral replace /*(t 
+ tf) by its definition from (1). This gives 

-lim ^ f^f{t + tf)dt f [fit - t) 

T-^CcTJ_j, Jq 

+ gif' — r)]dK{T) = — lim ^ f dKij) 
T^CO T Jq 

T 

if we reverse the order of integration. Since 
we assume that / and g are uncorrelated, how- 
ever, the product f (t + tj g (t — t) in this ex- 
pression makes no contribution to the final re- 
sult, and by replacing the integral of f{t + tj 
fit — t) by its value in terms of cf)^ the expres- 
sion as a whole can be written as 

— 2^ (f)i {tf t) dKir). 

The third integral in (6) can be simplified in 
similar fashion. The final result becomes 


with which we are confronted is evidently that 
of choosing K to make the mean square error 
as small as possible for any given <^’s. This 
problem will not be attacked here, although a 
solution obtained by a somewhat indirect 
method is presented in the next chapter. The 
principal reason for deriving equation (7) is 
to demonstrate the very important fact that 
the mean square error depends only upon the 
ttvo autocorrelations. No other characteristics 
of the input data need be considered. 

It will be recalled that the mean square cri- 
terion was introduced originally on the ground 
of mathematical convenience. This leaves un- 
settled the question of how good a measure of 
performance for a data-smoothing network it 
actually is. This is a critical question, since 
upon it depends the validity of the whole ap- 
proach outlined in this chapter. A priori, the 
least squares criterion is a dubious one since 
it gives principal weight to large errors. In 
fire control we are normally interested only in 
shots which are close enough to register as hits. 
If a shot misses it makes little difference 
whether the miss is large or small. The merits 
of the least squares criterion are considered 
in more detail in Chapter 9, where the conclu- 
sion is reached that the criterion is probably 
adequate for many problems but needs to be 
supplemented or replaced in others, including 
the special case of heavy antiaircraft fire to 
which particular attention is given in this 
monograph. Pending the discussion in Chapter 
9, the least squares criterion will be assumed 
to be a valid one, with the understanding that 
the analysis is intended primarily for its value 
in contributing to the general understanding of 
the data-smoothing problem rather than as a 
means of fixing the exact proportions of an op- 
timal smoothing network. 



= <^1 (0) - 2 / 01 ( ^/ + r) dKir) 

•/o 

^CO ^00 

-j- / dKir) I [0i(t — O') + 02(t — (r)]dA((r) . 

Jo J 0 

The only quantities appearing in equation 
(7) are the autocorrelations, 0i and 0o, of the 
true target path and the observational error, 
and the function K which specifies the data- 
smoothing structure. The theoretical problem 


7 6 DATA SMOOTHING AS A FILTER 
(7) PROBLEM 

The time-series approach to data smoothing 
is closely associated with another which at first 
sight may seem quite different. This second 
approach is suggested by the procedures used 
in communication engineering. Here the sig- 
nals, be they voice, music, television, or what 
not, are again time series. Instead of dealing 




CONFlDKN'ri 


80 


FORMULATION OF THE DATA-SMOOTHING PROBLEM 


with actual signals varying in a more or less 
irregular and random manner with time, how- 
ever, it is customary to deal with their equiva- 
lent steady-state components on the frequency 
spectrum.’’ 

The analysis of data smoothing can conven- 
iently be approached by supposing that both 
the true path of the target and the effects of 
tracking errors are represented, in a similar 
way, by their frequency spectra. When the 
situation is presented in this way, however, 
there is an obvious analogy between the prob- 
lem of smoothing the data to eliminate or re- 
duce the effect of tracking errors and the prob- 
lem of separating a signal from interfering 
noise in communication systems. We may take 
as an example of the latter the transmission 
of voice or music by ordinary radio over fairly 
long distances, so that the effects of static in- 
terference are appreciable. In such a system 
a reasonable separation of the desired signal 
from the static can be obtained by means of 
a filter. In a representative situation an ap- 
propriate filter might transmit frequencies up 
to perhaps 2,000 or 3,000 cycles per second,® 
while rejecting higher frequencies. 

The choice of any specific cutoff, such as 
2,000 or 3,000 c, in the radio system depends 
upon a compromise between conflicting consid- 
erations. Both speech or music and static nor- 
mally include components of all frequencies 
which can be heard by the human ear. Thus, 
suppressing any frequency range below the 
limits of audibility, at perhaps 10,000 or 20,000 
c, will injure the signal to some extent. The 
intensity of the signal components, however, 
diminishes rapidly above 2,000 or 3,000 c, while 
the energy of the static interference is more 
evenly distributed over the spectrum. Thus, by 
Altering out the first 2,000 or 3,000 c, we can 
retain most of the signal while rejecting most 
of the noise. Naturally, the exact dividing line 
will depend upon the relative levels of signal 
and noise power. If the static interference is 
quite weak, for example, it would be worth 

^ The review of communication theory given in Ap- 
pendix A shows how this equivalence is established by 
Fourier or Laplace transform methods. 

° In practice, of course, the filtering would probably 
take place in the radio-frequency circuits, but it is 
more convenient here to think of it occurring in the 
demodulated output. 


while to transmit a considerably wider band 
in order to retain a more nearly perfect signal. 
If the static level is extremely high, on the 
other hand, it would be necessary to transmit a 
still narrower band at the cost of greater mu- 
tilation of the signal. 

The separation of the true path of a target 
from the observed path including tracking 
errors, as a preliminary to prediction of the 
future position of the target, presents an ap- 
proximately analogous situation. Again the 
spectrum of the “signal’* or true path is con- 
centrated principally in a low-frequency band, 
in most instances, while the energy of tracking 
errors or “noise” appears principally at con- 
siderably higher frequencies. Thus the two can 
be separated by a low-pass filter. The separa- 
tion, however, is not complete since some com- 
ponents of the signal spectrum extend into the 
noise region. Thus the smoothing process must 
be accompanied by some mutilation of the sig- 
nal, and the optimum compromise is again 
attained from a filter which transmits a rela- 
tively broad band when the tracking errors are 
of low intensity and a much narrower band 
when they are large. 

In these terms the most obvious difference 
between the data-smoothing problem and the 
static interference problem in the radio system 
is in the order of magnitude of the frequencies 
involved. They are roughly 10,000 times smaller 
in the data-smoothing case. Thus, the typical 
signal band in a fire-control system may cover 
a few tenths of a cycle per second, in compari- 
son with a useful band of 2,000 or 3,000 c in a 
radio system, and the spectrum of tracking 
errors or noise, with representative tracking 
devices, includes appreciable components up to 
perhaps 2 or 3 c, in comparison with a total 
effective noise band in the radio system ex- 
tending to the limits of audibility at perhaps 
20,000 c. 

This analogy between data smoothing and 
the filtering problems which appear in ordi- 
nary communication systems transmitting 
speech or music must of course not be carried 
too far. For example, previous experience with 
communication filters is of no help in fixing in 
detail the cutoff in attenuation characteristic 
of the data-smoothing filter, since in communi- 
cation systems these choices depend on psycho- 


PHYSICAL AND TACTICAL CONSIDERATIONS 


81 


logical considerations of no relevance in the fire- 
control problem. Methods of determining the 
best rules for proportioning a data-smoothing 
filter, therefore, remain to be determined. We 
may also notice that, whereas the time-series 
approach was of the data-smoothing and pre- 
diction type, the filter approach emphasizes 
data smoothing only. The addition of the pre- 
diction function can be expected to change ma- 
terially the overall characteristics of the cir- 
cuit. Neither of these remarks, however, robs 
the filter approach of its value as a simple way 
of thinking about the problem qualitatively. 

7 7 RELATION BETWEEN TIME-SERIES 
AND FILTER APPROACHES 


the power per cycle. The relation is therefore 
equivalent to the statement that the autocorre- 
lation and power spectrum are Fourier trans- 
forms of each other. 

Since we have already established the fact 
that the mean square error in prediction de- 
pends only on the autocorrelation, this analysis 
enables us to conclude immediately that the 
mean square error can also be calculated from 
the power spectra of the signal and noise. It 
is entirely independent of the phase relations 
in either signal or noise. The phase character- 
istics of the data-smoothing network, which 
operates on the signal after a specific wave 
shape has been established, is, of course, still 
of consequence. 


The time-series and filter methods of looking 
at data smoothing are related to one another 
by the fact that the autocorrelation can be com- 
puted from the amplitude spectrum, or vice 
versa, by Fourier transform means. Consider, 
for example, the Fourier transform of the 
autocorrelation. If we make use in particular 
of (4) we have 

f <I>,N (T)e 


- WW,/" + 


\/ ‘Itt 

Wi 


FMFM 


(8) 


where 


f’M = KOe-'^^dt 

= f_^ f{i + r) dr . 


(9) 


F((d) is of course the steady-state spectrum 
of the signal fit). Equation (8) thus states 
that the Fourier transform of is equal to a 
constant times the square of the amplitude of 
the steady-state spectrum. The amplitude 
squared spectrum is, however, a measure of 


7 » PHYSICAL AND TACTICAL 

CONSIDERATIONS 

Thus far the material which has been pre- 
sented has been primarily mathematical. It 
has consisted, in other words, of outlines* of 
general analytical methods which are available 
for use with the data-smoothing problem. It is 
also possible to approach the problem in a 
much more concrete fashion. It is obvious that 
by giving thought to the details of the physical 
characteristics of tracking units and targets, 
and to the tactical situations with which we 
expect to deal, it should be possible to draw a 
number of specific conclusions about the prob- 
lem as a whole. In a general theory of the de- 
sign and tactical use of fire-control apparatus 
such an approach might well be a primary one. 
It is scarcely possible to follow it in detail in 
the present discussion. The following para- 
graphs, however, indicate some of the kinds of 
considerations which can be brought into the 
problem in this way. It will be seen that they 
tend to modify the strictly mathematical ap- 
proach, partly by qualifying to some extent the 
assumptions made in the mathematics, and 
partly by tending to give much more emphasis 
to particular aspects of the problem than would 
appear in a general analytic outline. 

Choice of Coordinates 

One of the most obvious omissions in the 
general analysis thus far is any consideration 
of the choice of coordinates in which the data 


rONFIDENTlA 3^ 


82 


FORMULATION OF THE DATA-SMOOTHING PROBLEM 


smoothing is to take place. So far as either 
the statistical or filter theory is concerned, the 
coordinates in the data smoother may repre- 
sent either the original tracking data or any 
transformation of them. The fact that there is 
actually something to be decided here, however, 
is easily seen from the long-range antiaircraft 
problem. The input tracking coordinates for 
antiaircraft would normally be azimuth, eleva- 
tion, and slant range. If the airplane flies in a 
straight line roughly overhead, the general 
shape of the azimuth and the azimuth rate as 
functions of time are given by the curves in 
Figure 2. The curves become indefinitely 



Figure 2. Azimuth and azimuth rate for crossing 
target. 


steeper as the target path approaches the 
zenith, and it will be seen that if the approach 
is reasonably close, either the azimuth or the 
azimuth rate must include a very substantial 
amount of high-frequency energy. Since the 
possibility of an effective separation between 
the signal and noise in the filter approach de- 
pends upon the assumption that the signal com- 
ponents are of quite low frequency with respect 
to the noise, the presence of this high-frequency 
energy is evidently serious. 

When the target describes a violently evasive 
path the signal spectrum must naturally in- 
clude substantial high-frequency components, 
whatever the coordinate system may be. The 
high-frequency components indicated in Figure 
2, however, are due to the fact that the target 
path happens to pass almost over the director 
and are essentially superimposed upon the 
high-frequency components which reflect the 
complexity of the target path itself. It is clear 


as a matter of principle that an acceptable 
coordinate system for data smoothing should 
not introduce frequency components which de- 
pend upon such accidental factors as the loca- 
tion and orientation of the coordinate system. 
The rectangular system mentioned in connec- 
tion with Figure 1 evidently meets this condi- 
tion; so also does the “intrinsic’" system de- 
scribed in the next section. 

Physical Limitations of Target or Tracker 

We may also approach the data-smoothing 
question by a consideration of the motions 
which are physically possible either in the 
target or in the tracking device. In the heavy 
antiaircraft problem, for example, there are 
substantial physical limitations on the per- 
formance possibilities of present-day aircraft. 
We can be quite sure that any motion incom- 
patible with these limitations is necessarily a 
tracking error and can be removed from the 
incoming data. Naturally, these limitations 
must appear in the power spectrum of the sig- 
nal if they affect the mean square error in pre- 
diction, so that their existence in no way dis- 
putes the mathematical framework we have 
set up. Consideration of the physical factors 
which produce them, however, may permit 
them to be established more easily or in more 
clear-cut fashion than would be possible from 
a statistical examination of target records 
alone. 

The limitations on airplane performance 
can be stated most simply when the motion of 
the airplane is expressed in so-called intrinsic 
coordinates. These are the speed of the air- 
plane, its heading, and its angle of dive or 
climb. The maneuvering possibilities of a con- 
ventional airplane in these three directions are 
quite unequal. By banking sharply it can 
maneuver violently to the right and left and 
thus make quick changes in heading. The pos- 
sibilities of maneuvering up and down, how- 
ever, are considerably less, particularly for a 
heavy airplane, where there are usually restric- 
tions on the maximum angle of dive or climb 
which can be assumed. The possibilities of 
quickly changing the speed of the airplane, 
finally, are almost nil. The thrust of an air- 
plane propeller is so small in comparison with 



PHYSICAL AND TACTICAL CONSIDERATIONS 


83 


the mass of the airplane that only small accel- 
erations are possible.^^ 

Thus the optimum filters for the three coor- 
dinates should be different. The one for speed 
can have a very narrow band, since most of 
the signal energy for this coordinate occurs at 
very low frequencies. The optimum band for 
the angle of dive or climb, however, should be 
larger (unless it turns out that pilots seldom 
make use of maneuvering possibilities in this 
direction) and the one for the heading larger 
still. In this ability to discriminate among the 
various possible directions of motion the in- 
trinsic coordinate system is evidently an im- 
provement even on the rectangular system. 


Settling Time 

Another aspect of the data-smoothing prob- 
lem which has not been given conspicuous at- 
tention in the purely mathematical discussion 
is the fact that in an actual tactical situation 
questions of elapsed time are of great impor- 
tance. Engagements usually begin suddenly 
and last for a comparatively brief period, and 
it is important to find a data-smoothing scheme 
which provides adequate firing data as quickly 
as possible after an engagement starts. A situ- 
ation essentially similar to the beginning of an 
engagement may also be presented whenever 
the target makes a sudden change of course or 
whenever it is necessary to shift from one 
target to another in a given attacking body. 
The time required for a computer to give 
usable output data after any of these events is 
its so-called “settling time,*' and is one of the 
most important parameters of any data- 
smoothing system. It is possible to make rough 
estimates of settling time by indirect means in 
both the statistical and filter theories of data 
smoothing, but no explicit consideration of 
necessary time lapses appears in either theory. 
Evidently, the fundamental fault lies with the 
“stationary** assumption. 


^ This ignores the possibility of changing the speed 
through gravitational forces. Since these possibilities 
are linked to the angle of dive or climb, however, they 
can be predicted. This has actually been done in one 
experimental computer. 


Effect of Human Factors 

Aside from the conditions on target perform- 
ance which arise from the physical character- 
istics of the target itself, there are others 
which are due to the fact that the target is 
under the control of a human being with a 
definite purpose. The language of the statistical 
and filter methods is broad enough to cover 
almost any situation. It tends to suggest, how- 
ever, that the typical target paths with which 
we deal are the relatively structureless conse- 
quences of random physical forces. The inter- 
vention of purposive human behavior, on the 
other hand, tends to give paths which fall into 
more or less definite patterns. A simple illus- 
tration is furnished by the argument which is 
frequently offered in defense of the straight 
line assumption in dealing with antiaircraft 
defense against heavy bombers. It is contended 
that while the targets may in fact engage in 
substantial evasive maneuvers during most of 
their flight, there will always be a substantial 
period during the bombing run in which they 
must fly very straight in order to achieve 
bombing accuracy. On the basis of ordinary 
probability we would of course expect substan- 
tial straight line segments quite infrequently 
if the course as a whole shows marked disper- 
sion, and the intervention of the human pilot 
thus provides a higher degree of structure than 
one would expect in a corresponding situation 
dominated by purely natural factors. 

A broader example is furnished by a com- 
parison of two airplanes, or perhaps more 
simply of two boats, one of which is under the 
control of a human operator, while in the other 
the steering- controls are lashed in a neutral 
position. Both boats, say, may be expected to 
experience small variations of course due to the 
random effects of wind and waves upon them. 
Over a short period of time the observed mo- 
tions of the two boats should be substantially 
identical. In the case of the boat with the 
lashed helm these random variations will tend 
to accumulate, so that it is possible to make a 
reasonable prediction of the position of the 
boat for only a comparatively short distance 
in the future. In the boat with the human 
steersman, on the other hand, we may expect 
corrections to be applied as soon as the random 
effects become large, so that the boat tends to 




[ONTIDEXTIAL 


f 


84 


FORMULATION OF THE DATA-SMOOTHING PROBLEM 


retain the same general course and it is pos- 
sible to predict its position hours or even days 
later from a relatively brief observation. 

Neither of these illustrations is inconsistent 
with the mathematical framework laid down 
earlier in the chapter, in a purely theoretical 
sense. For example, the bombing run illustra- 
tion merely states that because of the presence 
of the human operator there are definite phase 
relations in the input signal. As we have seen, 
such relations can exist without affecting com- 
putations based on mean square error. The 
comparison between the piloted and pilotless 
boats can be interpreted as the result primarily 
of differences in the signal power spectrum. 
In the case of the pilotless boat, for example, 
the signal occupies a fairly continuous low- 
frequency band, while in the case of the piloted 
boat it must be regarded as concentrated very 
closely around zero frequency, so that it is ap- 
proximately a line spectrum superimposed on 
a continuous one. The formal mathematical 
theory covers also such cases as these. 

The point of this discussion, however, is that 
the mathematical theory, although it is suf- 
ficiently general in a formal sense, fails to dif- 
ferentiate between such situations as those 
just described ,and the more shapeless sort 
involving continuous spectra with random 


phase relations, even if the special features in 
these situations may be the controlling factors 
in determining the actual probability of hit- 
ting. If we could believe the bombing run 
hypothesis, for example, and had a sufficiently 
accurate computer and gun, we could expect 
to score a hit in every engagement, no matter 
how large the mean square error might be. 
More generally, it is probably only the ten- 
dency of targets to exhibit “line spectra” which 
prevents the real probability of a kill, small 
at best, from becoming microscopic. It is nec- 
essary to lay special emphasis on these factors 
in order to keep the overall fire control picture 
in perspective. 


Criterion of Performance 

Last on this list of doubts about the statisti- 
cal and filter theories, we may mention the 
least squares criterion of accuracy. This was 
discussed before, but it is mentioned again as 
a matter of emphasis, and because of its close 
relation with the factors we have just dis- 
cussed. For example, the bombing run illustra- 
tion obviously represents one situation in 
which the mean square error is not a good 
guide to the actual probability of scoring a hit. 




Chapter 8 

STEADY-STATE ANALYSIS OF DATA SMOOTHING 


I T WAS SHOWN in the previous chapter that 
both the statistical and filter theory ways of 
looking at the data-smoothing problem lead 
naturally to an analysis in terms of the power 
spectra of the signal and noise. The phase rela- 
tions are not important as long as we accept 
the mean square error as a criterion of per- 
formance. The inadequacies of the mean square 
criterion will finally force us to abandon the 
steady-state attack in favor of a direct analysis 
in terms of the wave shapes of some assumed 
signals. The steady-state attack is nevertheless 
a very useful one. This chapter will conse- 
quently continue the analysis from this point 
of view. It will be assumed as heretofore that 
the heavy antiaircraft problem is the particular 
subject of interest. 

A large part of the discussion hinges upon 
the conditions which must be satisfied by the 
external characteristics of an electrical net- 
work if it is to be capable of physical realiza- 
tion in any way whatever. These limitations 
and the characteristics which may be postulated 
for physical networks are decisive since, in the 
absence of such restrictions, no limits could be 
set upon the performance which might be ex- 
pected from data-smoothing and predicting 
circuits. The facts about physically realizable 
networks which we shall find of most use are 
summarized below, but the reader not familiar 
with this field is urged to read also the account 
given in Sections A. 9 and A.IO, Appendix A.^®^ 
The conditions which must be satisfied by 
physically realizable networks can be stated in 
either transient or steady-state terms. In tran- 
sient terms they are expressed most simply by 
the statement that the response of a physical 
network to an impulsive force must be zero up 
to the time the force is applied. Thus the net- 
work has no power to predict a purely arbi- 
trary event. That is, it has no way of foresee- 
ing whether or not an impulse is actually going 
to be applied to it. This characteristic of physi- 
cal networks is taken as a postulate. 

The steady-state limitations on physical net- 


works are expressed in terms of their attenua- 
tion and phase characteristics. They may be 
derived either from the transient specification 
or from the postulate that a physical network 
must be stable. There are no important limita- 
tions to be placed upon the attenuation and 
phase characteristics of physical networks as 
long as we deal with these characteristics sepa- 
rately, but there are very severe limitations on 
the phase characteristic which can be associated 
with any given attenuation characteristic or 
vice versa. In particular, when the attenuation 
characteristic is prescribed, there is a definite 
formula for calculating the unique limiting 
phase characteristic with which it may be asso- 
ciated.^'^*' This is the so-called “minimum phase” 
characteristic because any other physical net- 
work having the postulated attenuation char- 
acteristic must have as great or greater phase 
shift at every frequency. As we shall see later, 
this greater phase characteristic would corre- 
spond to longer lags in obtaining usable data, 
so that the minimum phase characteristic is 
the optirnum for a data-smoothing network. 
The minimum phase characteristic has the addi- 
tional important property that not only does 
it specify the transfer admittance of a physical 
network, but the reciprocal of that transfer 
admittance can also be realized by a physical 
structure.^ 

In addition to this principal formula for the 
relation between attenuation and phase there 
are a number of subsidiary expressions for 
special aspects of the problem. One in partic- 
ular, relating the attenuation to the behavior 
of the phase characteristic in the neighborhood 
of zero frequency, is used extensively in this 
chapter. 


In limiting cases, such as may be found when the 
transfer admittance contains zeros or poles exactly on 
the real frequency axis, the “physical structure” may 
require such constituents as ideally nondissipative re- 
actances, perfect amplifiers with unlimited gain, etc. 
This, however, is of no consequence for the present 
general discussion. 


6q^FIDE]\TIAC' 


85 


86 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


8 1 THE SIGNAL SPECTRUM 

It is natural to begin with a discussion of the 
spectrum of a typical target path. Unfortu- 
nately no data on the spectra of actual meas- 
ured airplane paths exist, and the theoretical 
assumptions which may be made about paths 
of airplane targets are best discussed in the 
next chapter. This section consequently will be 
confined to rather general observations about 
the problem. It will be convenient to assume 
for definiteness that the quantities to be 
smoothed are the velocity components in Car- 
tesian coordinates. 

The simplest point of departure is furnished 
by the conventional assumption that the target 
flies in a straight line at constant speed. If we 
could construe this assumption literally, it 
would mean that the velocity spectrum in rec- 
tangular coordinates would reduce to a single 
line at zero frequency. In practice, of course, 
the spectrum is not so simple. Even in the 
absence of deliberate maneuvering, the target 
will fly a slightly curved path because of 
“wander.” Moreover, even if the target could 
fly exactly straight, the single line spectrum 
would apply only to a straight course in- 
definitely continued. The spectrum becomes 
more complicated if we consider the fact that 
tracking must have begun at some finite time 
in the past, or that the target may presumably 
change occasionally from one straight line 
course to another. 

As a result of both these causes, the actual 
signal spectrum must be regarded as occupying 
a band bordering on zero frequency. The distri- 
bution of energy in detail will, of course, 
depend on particular circumstances. The band 
has no very well defined upper limit, but in 
most cases the great bulk, at least, of the 
energy should be below, say, one-fourth or one- 
fifth of a cycle per second. For example, the 
natural periods of a heavy airplane, which one 
would expect to be correlated with wander, are 
below this limit.-^ This limit is also sufficient to 
include most of the energy resulting from 
changes in course occurring as frequently as 
every ten or twenty seconds. 

In general, it is to be supposed that the sig- 
nal spectrum varies as w"”, where n may be 
1, 2, 3, depending on the frequency range. This 
follows from general considerations of the 


limitations of airplane performance. Thus, if 
we suppose that the velocity changes discon- 
tinuously from time to time, it follows from 
general Fourier principles that the amplitude 
must vary as o>-L This is presumably a fair 
representation of the actual signal spectrum at 
low frequencies. At moderate frequencies, how- 
ever, we must take account of the fact that the 
velocity can actually be changed rapidly but 
not discontinuously, and we consequently 
assume that the amplitude begins to vary as 
0 )-^ Finally, at frequencies of the order of per- 
haps one cycle per second one must take ac- 
count of the fact that the airplane must bank 
in order to turn. Since it takes some time to roll 
into the bank, even the acceleration in the lat- 
eral direction cannot be discontinuous, and 
consequently the amplitude must begin to vary 
as The application of such successive limit- 
ing factors in constructing a complete spec- 
trum is described in more detail in Section A.8 
of Appendix A. 

One other general condition of the same kind 
can be mentioned. It can be shown'^ that the 
integral from zero to infinity of log H/1 + 
where H is the power spectrum, is very impor- 
tant in determining the properties of a time 
series. More explicitly, the integral converges 
if the series is essentially statistical, so that we 
cannot foretell the future from the past with 
absolute certainty. This of course is the case 
with an actual signal spectrum in a fire-control 
problem. It implies two consequences ; first, 
that H cannot be zero over any finite band ; and 
second, that in the neighborhood of infinite fre- 
quency H diminishes slowly enough so that 
I log H\/m 0. 

82 THE NOISE SPECTRUM 

The spectrum of tracking errors depends 
largely upon the particular sort of tracking 
equipment involved. Broadly speaking, optical 
tracking equipment (at least that of the present 
or recent past) tends to produce tracking errors 
not only of small amplitude, but also of low 
frequency, so that they are hard to separate 
from the signal spectrum. Radar equipment, of 
the present time, produces higher-frequency 
errors. Relatively high-frequency errors are 
particularly likely to be found in very stiff 
automatic tracking radars. 


RANDOM NOISE FUNCTIONS 


87 


A number of examples of spectra of tracking 
errors are shown in Figures 1, 2, and 3. The 
spectra are given directly in terms of range 
and angle errors. To make them comparable 
with the velocity spectra described previously 


“random noise’' functions.’’ A random noise can 
be defined as a function which has a definite 
amplitude spectrum but completely random 
phase characteristics. The theory of such func- 
tions is well developed because of their frequent 



Figure 1. Power spectrum of range errors of ex- 
perimental radar. 


it would be necessary to multiply all amplitudes 
by w. In addition, it would of course also be 
necessary to multiply the angle rates by some 
suitable range in order to compare them di- 
rectly with the yards-per-second rates we have 
otherwise considered. 

After multiplication by o>, the radar spectra 
appear to be about fiat up to perhaps one cycle. 
Beyond that point they no doubt drop off 
slowly, although the accuracy of the data is not 
sufficient to permit the situation to be stated 
very exactly. 

8”’ RANDOM NOISE FUNCTIONS 

The properties of the signal and noise as we 
shall assume them here can be conveniently 
expressed by reference to the theory of so-called 



Figure 2. Power spectrum of angular height 
errors of experimental radar. 


occurrence in physics. It is probable that 
neither our noise functions nor our signal func- 
tions are, strictly speaking, random noise ac- 
cording to this definition. Thus, there are proba- 
bly certain definite phase relations in our noise 
functions because of the physical character- 
istics of tracking devices. There is no evidence, 
however, that any such relations are important 
enough to be significant in the data-smoothing 
problem, so that we are fully justified in iden- 
tifying them with random noise functions as 
defined above. The phase relations in the signal 
are by no means random. As long as we con- 
sider only the mean square error, however, this 
factor is immaterial, and we can replace the 
actual signal by a random noise function with 
the same power spectrum for purposes of 
analysis. 

The most familiar example of a random 
noise function is furnished by the thermal 


The fact that we also refer to tracking errors as 
“noise” is, of course, merely a coincidence. 




88 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


voltage across a resistance R. This is a random 
noise whose spectrum is constant up to very 
high frequencies with the value P = AkTR (k 
is Boltzmann’s constant and T the absolute 
temperature) . A second example is black body 



POWER SPECTRUM 
TRAVERSE ERRORS 
RMS = I.4MIL 
MEDIAN =0.31 CPS 



PER CPS 

9 C 




to 

_J 

/V 




— 6 

01 

LU 

2 A 




/ 




/ UJ 

/ ^ 


0 2 4 6 8 10 12 14 16 

FREQUENCY IN UNITS OF —CPS 
45 


Figure 3. Power spectrum of traverse angular 
errors of experimental radar. 


radiation. If there is black body radiation in a 
space, the electric (or magnetic) field intensity 
at a point is a random noise function with 
spectrum 


P{f) = 


Stt/^ 1 

(73 ^hf/M _ 


according to Planck’s law. Random noise func- 
tions also occur in the Schottky effect, in 
Brownian motion, and in diffusion and heat 
flow problems. 

For purposes of analysis, a random noise 
function can be thought of as a function made 
up of a large number of sinusoidal components, 
which are very closely spaced in frequency 
and whose phases are completely ran- 
dom.2^-23a Thus a random noise can be repre- 
sented as 


N 

^ a„ cos (cOnf + <^n) 

n — \ 

where <o„ = tiA/, A/ being the frequency differ- 
ence between adjacent components. The phase 


angles are random variables which are in- 
dependent with a uniform probability distribu- 
tion from 0 to 27r. As A/ decreases the functions 
in this ensemble approach, in a certain sense, 
a limiting ensemble, providing the amplitudes 
an are adjusted properly. What is desired is to 
have the total power in the neighborhood of 
each frequency approach a certain limit P(/), 
the power spectrum at that frequency. To do 
this we make 

a„2 = 27rP(/)A/. 

In the limiting ensemble the total power within 
a small frequency range A/ is then P(/)A/. 
The function P(oi) completely describes the 
random noise ensemble from the statistical 
point of view. 

A particularly important special case is that 
of a random noise with a constant power spec- 
trum. This is often called ‘‘flat” or “white” 
noise. True constancy out to infinite frequencies 
is of course impossible since it would imply an 
infinite total power in the function. The idea 
is, however, still useful and can be approxi- 
mated, as with resistance noise, by having a 
spectrum which is constant out to such high 
frequencies that behavior beyond this point is 
of no importance to the problem. We may con- 
veniently think of flat random noise as being 
made up of a succession of weak impulses oc- 
curring frequently but at random times with 
respect to one another. This results from the 
fact that a Fourier analysis of a single impulse 
gives a flat spectrum, and the random occur- 
rence of many of them produces a random set 
of phases. In a physical problem, such as resis- 
tance noise or Brownian motion, these im- 
pulses might correspond to the effects of indi- 
vidual small particles. Such a situation is of 
course completely chaotic. If the impulses are 
large and occur relatively infrequently, the 
power spectrum is still flat, though the func- 
tion is no longer a random noise function as 
defined here. This conception, which corre- 
sponds to a physical situation including definite 
causative elements, will be revived later under 
the name of the elementary pulse method of 
analysis. 

Random noise functions have a number of 
interesting characteristics. For example, they 
have the “ergodic property.” This means that 


20NFIDENTIA] 


PROPORTIONS FOR A DATA-SMOOTHING FILTER 


89 


averaging a statistic along the length of a par- 
ticular random function gives the same results 
as averaging the same statistic over an 
ensemble of functions having the same power 
spectrum. Each function is typical of the 
ensemble. To be more precise one must admit 
exceptions, but the probability of an exception 
is zero. For example, if we determine the frac- 
tion of time a given random function f(t) has 
a value greater than some constant A, it will 
be equal to the fraction of all functions in the 
ensemble which are greater than A at ^ = 0 
(with probability 1). 

A second characteristic of random noise 
functions is the fact that they frequently lead 
to Gaussian or normal law distributions. For 
example, the amplitudes of a random noise 
function are distributed about zero in accord- 
ance with the normal error law. Likewise, the 
amplitudes for two points spaced a given dis- 
tance apart form a two-dimensional normal 
error law distribution when we consider all 
possible positions of the first point. It is ap- 
parent that if the signal and noise are actually 
random functions the mean square error is as 
good a criterion of performance as any other, 
since it completely fixes the distribution in a 
normal law case. 

A final property of random noise functions 
is the fact that if a random noise is passed 
through a filter the output is still a random 
noise. If the power spectrum of the noise is 
P((o) and the transfer characteristic of the 
filter is y(i(o), the output spectrum is 
P(w) |F(iw) |2. In particular, if we take the 
derivative of a random noise with spectrum 
P((o) we obtain one with spectrum w^PCw). 

This last property of random noise functions 
suggests a method of representing them which 
we shall find useful in the future. The method 
is represented by Figure 4. It consists of a 


FLAT 


SHAPING 

NOISE 

SOURCE 


FILTER 


Figure *4. Circuit representation of random 
functions. 

source of flat noise followed by a shaping filter 
to give the desired power spectrum. We can 
easily assign to the filter the characteristics of 
a physically realizable structure by making use 


of the relations between attenuation and phase 
mentioned earlier in the chapter. It is merely 
necessary to convert the desired power spec- 
trum into a specification of the attenuation 
characteristic of the filter and then use the 
loss-phase formula to compute the correspond- 
ing phase shift. It will be assumed that this 
procedure has been followed when we make use 
of this circuit at a later point. 

The method of representing random func- 
tions shown by Figure 4 illustrates graphically 
the basis of the prediction schemes described 
thus far. The flat noise is of course absolutely 
unpredictable. The history of the function up 
to any given instant gives no indication of its 
value even a microsecond later. The filter, how- 
ever, forces the output current to have a cer- 
tain structure on which a prediction may be 
based. For example, if the filter will pass only 
very low frequencies it is clear that the output 
can change very little in a microsecond. 

8 4 THEORETICAL PROPORTIONS FOR 
A DATA-SMOOTHING FILTER 

The signal and noise spectra furnish the raw 
material from which a suitable data-smoothing 
filter can be deduced. We have still to deter- 
mine, however, the exact rule for choosing the 
cutoff and attenuation characteristic of the 
filter from these spectra. It is clear that previ- 
ous experience with signal-to-noise problems 
in systems transmitting voice or music is no 
help, since the filter proportions here depend 
upon psychological considerations of no rele- 
vance to the fire-control problem. For example, 
the interfering effect of a small amount of 
noise is much greater than one might expect 
from energy considerations, especially in in- 
tervals of low message level, and it is con- 
sequently worth while to maintain a relatively 
high level of attenuation in the noise band. 
Conversely, the breadth of the band required 
for the message depends as much on the ability 
of the ear to reconstruct a complete signal 
from an incomplete one as it does upon the 
actual signal power spectrum. 

In the data-smoothing case a suitable crite- 
rion, dependent upon more physical considera- 
tions, can be obtained by minimizing the rms 
error at the filter output. This criterion is 


rCONFlDENTI®! 


90 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


easily developed from the power spectrum ap- 
proach, and in a sense it is, of course, the only 
possible one as long as we follow the methods 
developed thus far. 

A very general theory for the minimization 
of the rms error of the filter output has been 
developed by Wiener.^ Since the power spec- 
trum approach is not the one we shall eventu- 
ally follow, however, it is not necessary to give 
this analysis in detail. The nature of the rela- 
tionships can be seen from an elementary com- 
putation. Thus in Figure 5 let OA be a unit 



Figure 5. Vector relation between input and out- 
put of data-smoothing network. 

vector representing the signal component at 
some particular frequency. Let the amplitude 
ratio between the input and output of the data- 
smoothing filter be x, and let it be assumed that 
the system is phase distortionless. This can 
always be accomplished, at the cost of lag, by 
phase equalization. Then the actual signal 
output can be represented by OJ5, where 
OB/OA = X. Let the ratio of noise power to 
signal power at this frequency be Then the 
output noise can be represented by the vector 
BCj at some arbitrary phase angle 6, where 
BC/OA = kx. 

The error in the output of the data-smooth- 
ing filter is evidently represented by the vector 
AC. have 

{ACy = (OA)H(l — X — kx cos sy -f- {kx sin oy] 
= (OA)2 [(1 — x^) — 2kx{l — x) cos 6 + k^x^]. 

Since 6 is random the cross-product term in- 
volving cos d disappears on the average. (More 
generally, it disappears as long as the noise and 
signal are uncorrelated, whether or not their 
relative phases are entirely random.) This 
leaves the mean square error as 

{AC) ^mean — ( 0^)2 [1 -2x A {I A k^)x^]. ( 1 ) 


The mean square error is a minimum if 


1 ^ Ps 

1 -f- A:2 p ^ p g 


where Ps and Pn are, respectively, the signal 
and noise power at this frequency. Upon sub- 
stituting this^ result in equation (1) and re- 
membering that (OA)^ = Pg, we find that the 
minimum mean square error is 


{Acy 


mean min 


PnPs 
Pn + Ps 


( 2 ) 


Equation (2) evidently represents the sought- 
for rule for the filter transmission character- 
istic. It is illustrated in Figure 6, where Ps 



Figure 6. Optimum transmission characteristic 
for data smoothing assuming signals with random 
noise characteristics. 



Figure 7. Signal and noise power spectra assumed 
in Figure 6. 


and Ps have been chosen respectively as the 
flat curve and the l/a)=* curve in Figure 7. In 
comparison with the characteristics of typi- 
cal filters in communication systems it is quite 


LAGS IN DATA-SMOOTHING FILTERS 


91 


rounded with a relatively slowly falling ampli- 
tude characteristic. More important than the 
detailed rule for. the transmission character- 
istic, however, is the conclusion that the shape 
of the characteristic is not very critical. There 
is very little loss in replacing the actual curve 
in Figure 6, by any other similar character- 
istic. For example, we might validate the 
assumption of zero phase distortion by making 
use of the curve which automatically gives a 
linear phase shift.^®® 

A more extreme illustration is furnished by 
the infinitely selective filter characteristic, with 
perfect transmission in the range in which the 
signal power is greater than the noise power, 
and zero transmission elsewhere, indicated by 
the broken lines in Figure 6. 

It follows from equation (1) that in the 
neighborhood of the cutoff point wo the mean 
square error for this filter is twice that of the 
optimum structure. In most frequency ranges, 
however, the penalty is far less than this. Since 
even a two-to-one change in the mean square 
error would produce no tremendous improve- 
ment in the effectiveness of fire, it is clear that 
the result to which we are led by this method 
of attack is by no means critical. 

« * LAGS IN DATA-SMOOTHING FILTERS 

The analysis just concluded has been directed 
at the amplitude characteristics of a data- 
smoothing filter. By virtue of the relations be- 
tween the amplitude and phase characteristics 
of physical networks mentioned earlier in the 
chapter, however, the analysis permits us to 



Figure 8. Some filter attenuation characteristics. 


give at least a partial description also of the 
phase characteristics of the filters. This is an 
important consideration because it bears upon 
the question of time delays in data-smoothing 
systems which was mentioned in Chapter 7. 

The general nature of the relationship in 
simple cases is illustrated by Figures 8 and 9. 



Figure 9. Corresponding minimum phase char- 
acteristics. 


Figure 8 shows a series of rising attenuation 
characteristics equivalent to rather unselective 
falling amplitude characteristics of the general 
type shown by the principal curve in Figure 6. 
Figure 9 shows the corresponding phase char- 
acteristics computed on a minimum phase shift 
basis. In Figure 8 the central attenuation char- 
acteristic B has been so chosen that the corre- 
sponding phase characteristic in Figure 9 is 
exactly a straight line at low frequencies, 
where the transmitted amplitudes are appreci- 
able. Curves A and C in the two drawings show 
slightly different cases, but it is clear from 
the figures that the tendency of the phase 
characteristics to approximate linearity is still 
marked. 

In communication engineering a phase char- 
acteristic proportional to frequency is inter- 
preted as indicating a delay in seconds equal to 
the slope dB/d(o of the phase characteristic. 
This relation is illustrated most simply by an 
ideal line. The ideal line has zero attenuation 
combined with a phase shift which is propor- 
tional to frequency and which at any given fre- 
quency is also proportional to the length of the 
line in question. If we apply any arbitrary 
wave to the line it is propagated down the line 
with a definite velocity and unchanged wave 
form. The time required for the wave to reach 


^CONFlDENTIAli 


92 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


any point on the line is equal to the slope of the 
phase characteristic to that point. 

In a structure like a filter, which has an at- 
tenuation characteristic varying with fre- 
quency, it is of course no longer possible to 
transmit an arbitrarily impressed wave with- 
out change in wave shape. Even if the applied 
wave is merely a suddenly applied d-c voltage 
or single frequency sinusoid, there is a tran- 
sient period before the response approximates 
its final value. In structures having a substan- 
tially linear phase characteristic over any fre- 
quency range in which they exhibit an appreci- 
able amplitude response, however, this total 
transient characteristic falls naturally into two 
parts. The first is a waiting period equal to the 
slope of the phase characteristic, during which 
the response is very small, whereas the second 
is a true transient period in which the response 
is substantial but does not resemble the final 
steady-state response. This is illustrated by 
Figure 10 which shows the voltage at the fifth 



Figure 10. Voltage at fifth section of conventional 
low-pass filter in response to unit d-c voltage. 


section of a conventional low-pass filter in 
response to a d-c voltage applied at zero time 
at the input terminals.^^ The end of the waiting 
period, as deduced from the slope of the phase 
characteristic, is indicated by the broken line. 

Delays of the sort just illustrated must be 
expected in a data-smoothing filter whenever 
the nature of the signal is changed. This hap- 
pens at the beginning of tracking, in changing 
from one target to another, or even in follow- 
ing a single target when the target makes an 
abrupt change in course. Since usable data in 
a fire-control system must be quite accurate, 
the delay to be allowed for must include both 
the initial waiting period and the subsequent 


transient period until the transient ripples 
have almost vanished. A considerable part of 
the art of designing data-smoothing networks 
consists in controlling the design so that these 
final transient ripples decay relatively rapidly. 
We are not yet ready to discuss this problem. 
It will turn out, however, that the minimum 
interval which can be assigned to the “true 
transient" period is about equal to that which 
must be allowed for the initial waiting period.*" 
Thus the slope of the phase characteristic can 
be used as an index of the lags which must be 
expected in data smoothing merely by doubling 
the delay to which the slope would normally be 
said to correspond. 

When we use the phase slope as an index of 
delay it becomes immediately apparent that 
lags are the necessary consequence of smooth- 
ing in physical circuits. This is easily seen by 
reference to the relations which must exist be- 
tween attenuation and phase characteristics in 
physical structures. An example is provided by 
the formula^®*^ 



(A- Ao)d 



(3) 

where A is attenuation, is the attenuation 
at zero frequency, and B is phase shift. In other 
words, the delay (measured by the slope of the 
phase characteristic at zero frequency) is pro- 
portional to the integral of the attenuation on 
an inverse frequency scale when the attenua- 
tion at zero frequency is taken as the reference. 
The equation thus states that the system will 
exhibit a lagging response as long as there is a 
net high-frequency attenuation. As a numerical 
illustration, let it be supposed that A is zero 
below (0 = 1. This corresponds to the estimate 
made earlier in the chapter that the input sig- 
nal components in antiaircraft work lie roughly 
in the ban*d below about 0.1 or 0.2 cycle per sec- 
ond. Let it be supposed also that A at higher 
frequencies is equal to 3 nepers, corresponding 
to an average amplitude reduction of about 20 


® This is not intended to imply that the distinction 
between the initial waiting period and the “true tran- 
sient” period is quite as sharp as it is in Figure 10. The 
selectivity in a data-smoothing filter is usually not 
great enough to justify the assumption that components 
beyond the linear phase region are of negligible im- 
portance. 


wiener’s prediction theory ZERO NOISE CASE 


93 


to 1. Then dB/da. at the origin is given from 
equation (3) as G/tt seconds, and in accordance 
with the rule just enunciated the minimum de- 
lay to be expected from such a structure in a 
data-smoothing application would consequently 
be 12/7r seconds. 

Aside from such specific quantitative rela- 
tions equation (3) is useful as a basis for a 
number of important qualitative conclusions. 
One, for example, is the fact that although a 
lag is a necessary concomitant of any system 
showing a high-frequency attenuation, the 
amount of the lag depends greatly upon the 
portion of the frequency spectrum in which 
the attenuation is found. Since the integral is 
taken on an inverse frequency scale, a small 
attenuation at low frequencies is much more 
important than a considerably greater attenua- 
tion further out in the spectrum. This points to 
the desirability of designing tracking instru- 
ments which generate principally high-fre- 
quency noise, even if the amplitude of the noise 
is somewhat increased thereby. We may also 
notice that since the attenuation is a logarith- 
mic function of amplitude an initial moderate 
reduction in the amplitude of disturbing noise 
may be much less expensive in lag than subse- 
quent attempts at further reduction. For ex- 
ample, an amplitude reduction from 100 to 10 
per cent over a given portion of the frequency 
spectrum produces no more lag than a subse- 
quent reduction from 10 to 1 per cent. 

WIENER’S PREDICTION THEORY- 
ZERO NOISE CASE 

In Chapter 7 we distinguished between what 
we called the simple data-smoothing problem 
and the data-smoothing and prediction prob- 
lem. The simple problem, with which this re- 
port is chiefly concerned, is the one which has 
been given principal attention thus far. On 
account of its broad interest, however, it seems 
worth while to include also a brief statement 
of Wiener’s solution of the general problem. 
The method of development used here is intui- 
tive and nonrigorous in comparison with 
Wiener’s own development, but it permits the 
principal relations to be established by very 
elementary means. 

It is convenient to consider first the zero 
noise case. The past history of the signal, then. 


is known perfectly, and the existence of a 
prediction problem depends entirely upon the 
fact that since the signal is assumed to be sta- 
tistical in character, its future is not com- 
pletely determined from its past. The situation 
can be thought of in the terms suggested by 
Figure 11. The actual signal output appears at 


FLAT 

Moi ^ r 


SHAPING 


NETWORK 

N? 


PREDICTING 

nui 0 L 
SOURCE 


NETWORK 
Nj 

P. 

p; 

NETWORK 
—Nj 


Figure 11. Schematic representation of Wiener’s 
prediction theory when there is no noise. 


Pj. In accordance with the discussion earlier 
in the chapter, we imagine this signal to be 
generated by passing flat noise through the 
shaping network N^. The transfer admittance 
Fi(tw) of is determined from the power 
spectrum of the signal by the procedure out- 
lined earlier and is a minimum phase shift char- 
acteristic. It will be recalled that minimum 
phase shift transfer admittances have the im- 
portant property that their reciprocals are also 
the transfer admittances of physically realiz- 
able networks. 

From Fi we can readily compute the tran- 
sient response characteristic of We shall 
assume for illustrative purposes that the im- 
pulsive admittance of takes the special 
shape shown by Figure 12. 







w,(t)/ 











0 2 4 6 8 10 ^ 


Figure 12. Assumed impulsive admittance of 
shaping filter. 

The flat noise is thought of as consisting of 
a large number of elementary impulses with 
random amplitudes and occurring at random 
times. For the purposes of this analysis, how- 
ever, it is sufficient to consider only the three 
unit impulses shown in Figure 13. Impulse B 
is supposed to occur at the instant at which 


4 (^ONFlDEHTlJttrl^ 


94 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


the prediction is to be made, A occurs two sec- 
onds in the past, and C, one second in the 
future. The response of to these three im- 
pulses will evidently be three curves of the 
sort given by Figure 12, suitably displaced in 
time as shown by Figure 14. 

A B C 


- 2-101 

Figure 13. Impulses giving rise to applied signal 
through shaping filter. 

The desired output of the predicting network 
is the curve of Figure 14 advanced by the pre- 
diction time, which we can assume, for illus- 
tration, to be two seconds. It may be assumed 



















SUM \ 













\ 

% • 

* 1 . 

1 1 \ 






x • / 

» •* 

$ 

i 

1 

A A 
/V ' 

t \ 

, \ 

\ 

\ 

\ 

^ \ 

\ \ 




1 i 

/ • 
/ ' 

/ / 

f 


' % 

N 

V 




L V / 

-2 0 2 4 6 8 


Figure 14. Applied signal at Pi. 


for the sake of preliminary analysis that the 
input of the predicting network is the three 
original impulses of Figure 13. The terminal 
P 2 at which they are supposed to appear is of 
course a purely fictitious one and is not acces- 
sible to us physically. We can, however, con- 
struct the equivalent terminal P ^ by imposing 
the actual signal from terminal on the net- 
work Ng, whose transfer admittance is the 
reciprocal of that of N^. 


Let the predicting network connected to ter^ 
minal P '2 be represented by N^. Obviously a 
perfect prediction would be secured if could 
be assigned the impulsive admittance shown in 
Figure 15, that is, an impulsive admittance 



Figure 15. laeal impulsive admittance of predic- 
tion network Na in Figure 11. 


equal to the impulsive admittance of the origi- 
nal network but moved forward by the 2-second 
prediction time. Then all the constituent curves 
and the sum curve in Figure 14 would similarly 
be moved forward. Of course we cannot assign 
iVg an impulsive admittance which is different 
from zero at negative times without postulat- 
ing a nonphysical network. It is, however, per- 
fectly possible to define from the portion of 
the impulsive admittance characteristic at posi- 
tive times, with the remainder set equal to 
zero. This gives an impulsive admittance of 
the type shown by Figure 16. When energized 
by the three unitary impulses, it gives the 
result shown in Figure 17. The contributions 
of impulses A and B are not affected by the 
absence of a negative time portion of the im- 
pulsive admittance, but the contribution of im- 
pulse C is lost. 

To formulate a physical prediction network 



Figure 16. Eealizable portion of required im- 
pulsive admittance. 


lONFIDENTIAL 




wiener’s theory GENERAL CASE 


95 


we have merely to find by conventional meth- 
ods the steady-state admittance corre- 
sponding to the impulsive admittance of Figure 
16. The two networks and may then be 



-2 0 2 4 6 8 

Figure 17. Response of realizable prediction net- 
work. 


combined to give a single structure with the 
transfer admittance ¥^¥2 = ¥J¥^ which will 
give the complete prediction when energized by 
the actual signal. 

The mean square error in prediction is 
easily determined from the fact that the con- 
tributions of all impulses of the sort repre- 
sented by C, occurring in the prediction in- 
terval, are lost. Since impulses in the fiat noise 
source occur at random times the mean square 

error is proportional to J (r) dr, where a 

is the prediction time and W is the impulsive 
admittance of Figure 16. Since the fiat noise 
impulses occurring after the time at which the 
prediction is made are surely unpredictable, it 
is clear that this error is the least we could 
expect any physical prediction network to have. 

« 7 WIENER’S THEORY— GENERAL CASE 
When the input data includes noise as well as 
the signal it is natural to think of the situation 


in the manner shown by Figure 18. The first 
source of fiat noise, together with the shaping 
network Na, is the combination we have already 
used to represent the signal in the noise-free 



Figure 18. Circuit representation of random func- 
tions representing signal and noise. 

case. The addition of noise is represented by 
the second independent source of fiat noise with 
its associated shaping network Ni,. They com- 
bine to give the total input measured at Pj. 

This diagram emphasizes the fact that we 
think of the noise and signal as originating 
from different physical sources. By postulate, 
however, we are not able to separate the 
sources experimentally. So far as any observed 
result is concerned, consequently, we may as 
well deal with the simplified structure shown 
in Figure 19 which contains a single source of 



Figure 19. Schematic representation of Wiener’s 
prediction theory when there is noise. 


flat noise and a single shaping network. The 
transfer admittance of the shaping network N, 
is determined by adding the power spectra of 
signal and noise, converting the result to an 
amplitude characteristic, and computing the 
corresponding minimum phase according to 
the methods already used for the noise-free 
case.^^ 

Although we cannot separate the signal from 


^ Note that the shaping network thus obtained is not 
the same as the one we would secure by adding the 
transfer admittances of Na and N„ in Figure 18 di- 
rectly. In order to realize the same total power at Pi 
in each case, it is necessary to begin by adding the 
powers rather than the amplitude characteristics asso- 
ciated with the two paths. 


96 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


the noise completely, we saw earlier that the 
mean square difference between the total input 
and the signal is minimized if we multiply the 
amplitude of the input at each frequency by 
the ratio of the signal power to the sum of the 
signal and noise powers. A fictitious filter 
having the prescribed amplitude characteristic 
is represented by in Figure 19 . We assigned 
a zero phase characteristic so that there 
may be no lag in producing the result at P3. 
Thus the output at at any instant represents 
the best conceivable estimate (in the least 
squares sense) of the signal at that instant. 
The assumption of zero phase, of course, makes 
nonphysical, since it must have at least the 
minimum phase characteristic associated with 
its prescribed amplitude characteristic. This, 
however, is not an objection here since the 
structure is introduced purely for purposes of 
analysis. 

The situation is now reduced to a form in 
which it is substantially equivalent to the one 
appearing in the zero-noise case. We assume a 
series of random impulses at P2 which would 
produce responses at P3. The problem is that 
of advancing the response to each impulse so 
that the same result appears a seconds earlier 
at terminal P^. The solution is represented by 
networks and N^, which discharge functions 
similar to those of the correspondingly labeled 
networks in Figure 11 . Thus, the network No 
is the reciprocal of and is provided to make 
terminal P\ equivalent to P2 as a source of im- 
pulses. Network is defined by an impulsive 
admittance obtained from the impulsive admit- 
tance between P2 and P3 by advancing the 
latter characteristic a units in time and then 
discarding the portion at negative time. 

In this procedure there is only one point at 
which the situation differs from that without 
noise. In the noise-free case, the original im- 
pulsive admittance which we wished to advance 
in time was identically zero at negative times. 
In order to secure a physically realizable re- 
sult, we needed only to discard the portion of the 
impulsive admittance between t — 0 and t = a. 
In the present situation, on the other hand, the 
impulsive admittance is taken from a path in- 
cluding the nonphysical network N^. Thus the 
admittance may be expected to take such form 
as that shown in Figure 20, with nonzero am- 


plitudes at both negative and positive times, 
and in order to secure a physical final network 
it is necessary to discard everything to the left 
of the line a. 



Figure 20. Typical impulsive admittance of best 

smoothing network N 4 in Figure 19. 

This difference in the impulsive admittance 
characteristics has two consequences. The first 
is the fact that since the uncertainty of the 
prediction is measured by the amount of im- 
pulsive admittance which must be discarded, 
it is evidently greater in the present case where 
we are discarding much more. The second is 
the fact that in the noise-free case uncertainty 
exists only for a positive prediction time. A 
negative prediction time, which corresponds, of 
course, to the determination of the value as- 
sumed by the signal at some time in the past, 
can be set into the analysis as easily as a posi- 
tive prediction time, merely by shifting the im- 
pulsive admittance to the right rather than the 
left. In the noise-free case, however, there is 
nothing to be discarded when we shift to the 
right, since the impulsive admittance with 
which we begin is in any case identically zero 
for negative times. Thus the uncertainty in. 
the determination of any past value of the sig- 
nal is zero. Since we have postulated no noise 
to confuse the data, this is, of course, an 
inevitable result. As soon as noise is included, 
on the other hand, there is no such sharp dis- 
tinction between the future and the past.® The 
uncertainty in the determination of the true 
value of the signal in the near past is almost 
as great as it is in estimating what the signal 
will be in the near future. As we go further 


® This statement is to be understood in a physical 
rather than a mathematical sense. It is not intended 
to imply that there may not be sharp changes of be- 
havior in the impulsive admittance at zero. 


'CONFIDENTIAL 



OVERALL CHARACTERISTICS OF PREDICTING NETWORKS 


97 


and further into the past the uncertainty 
gradually diminishes. If we can allow ourselves 
unlimited lag, we at length reach a point at 
which the discarded portion of the impulsive 
admittance characteristic is negligibly small. 
This, however, does not mean that all uncer- 
tainties have disappeared, but merely that we 
can base our estimate of the signal upon the 
power-ratio rule developed previously. 


8 « OVERALL CHARACTERISTICS OF 
PREDICTING NETWORKS 

It has been fairly easy to develop a qualita- 
tive picture of the general characteristics of 
typical data-smoothing networks. As we have 
seen, they have amplitude characteristics of the 
low-pass filter type combined with lagging 
phase shifts. No corresponding qualitative pic- 
ture of the characteristics of a typical overall 
predicting circuit has, however, been developed 
as yet. The discussion just concluded provides 
a rule for determining the characteristics of a 
predicting circuit in any given case, but pro- 
vides comparatively little in the nature of a 
description of the result we may expect to 
secure. 

In any particular situation we can, of course, 
calculate the overall characteristics of the pre- 
dicting circuit. A simpler way of character- 
izing the overall predictor characteristic quali- 
tatively, however, is based upon the use of the 
attenuation-phase relations for physical net- 
works. We need merely use such an equation 
as (3) backward. Thus, we have previously 
shown that a positive phase slope corresponds 
to a lagging output. Correspondingly, a nega- 
tive phase slope can be interpreted to repre- 
sent a lead, or in other words, a prediction.^ 


^ This, of course, does not mean that a network with 
a negative phase slope can predict a perfectly arbitrary 
event. We can hope to realize a negative phase slope, 
in combination with a flat amplitude characteristic, 
over only a finite band. The spectrum of an arbitrary 
event, that is, any suddenly applied signal, will always 
include important components running out to infinite 
frequency, where the negative phase slope can no longer 
be realized. The statement does, however, mean that if 
we suddenly apply a signal made up of one or more 
low-frequency sinusoids, and wait for the steady state 
to become established, the output will appear to lead 
the input by a time equal to the slope of the negative 
phase characteristic. 


If we assign {dB/dM)^-Q in equation (3) a 
negative value, we see that A—A^ must on the 
average be negative. In other words, the am- 
plitude characteristic of an overall prediction 
circuit must rise, on the average, as we proceed 
upward from zero frequency. This is in marked 
contrast to a data-smoothing network, which, 
as we have seen, tends to have a low-pass filter 
type of characteristic with a falling amplitude 
characteristic at high frequencies. The in- 
creased amplitude of response may have two 
detrimental effects. In the first place, it evi- 
dently produces a distorting effect on any sig- 
nal components to which it applies. In the 
second place, it produces an exaggerated re- 
sponse to noise. 

Examples of the characteristics of overall 
prediction circuits are readily constructed by 
reference to the circuit of Figure 21. Various 



Figure 21. One-dimensional prediction circuit 

with data-smoothing networks. 

particular results are obtained by assigning 
particular characteristics to the data-smooth- 
ing network. Thus, if the data-smoothing net- 
work is absent entirely the transmission 
through the path containing the differentiator 
is iiiitf, since differentiation is equivalent to 
multiplication by iw. The attenuation of the 
overall circuit is consequently A = — log 
|1 + This is plotted as curve I of Figure 
22. The increasing amplitude characteristic at 
high frequencies is obviously due fundamen- 
tally to the increased transmission through the 
differentiator circuit. 

If the data-smoothing network is assigned 
the characteristic (1 + ta)a)"b corresponding to 
a very simple low-pass filter type of response, 
the overall transmission becomes that shown 
by curve II in Figure 22. (It is assumed that 
a = tf, for simplicity.) The negative attenuation 
at high frequencies is much reduced. This is 
paid for by an increased amplitude of response 
at low frequencies, but since the integration in 
(3) takes place on an inverse frequency scale, 
the low-frequency fragment is much less than 
the gain reduction at high frequencies. Curve 


/ctxfidextial 




98 


STEADY-STATE ANALYSIS OF DATA SMOOTHING 


III shows the result when the data-smoothing 
network is assigned the characteristic 
(1 + Finally, curve IV shows the result 

obtainable when there is also a filter in the 



Figure 22. Attenuation characteristics of predic- 
tion circuit shown in Figure 21. 

present-position circuit (as shown by the 
broken lines in Figure 21), so that there may 
be a net positive attenuation at high fre- 
quencies. 

In view of the inverse frequency scale in (3) , 
the gross negative attenuation will be mini- 
mized if the negative attenuation region is 
placed very close to zero frequency. This, how- 
ever, means that much of the signal energy 
falls in the negative attenuation region so that 
in certain respects, at least, the signal response 
must be seriously injured. For example, in the 
specific circuits just discussed we can place the 
negative attenuation region at very low fre- 
quencies by choosing very long time constants, 
a, in the data-smoothing networks, with the 
consequence that the circuits will operate cor- 
rectly for any long continued straight line path, 
but will be very sluggish in changing from one 
straight line to another. If the negative attenu- 
ation region is placed at higher frequencies, on 
the other hand, the signal response is improved 
but beyond certain limits the circuit becomes 
unbearably sensitive to noise. 

Quantitative illustrations of these relation- 
ships are quickly constructed. Suppose, for ex- 
ample, that the prediction time is 2 seconds. 
From (3) this is consistent with an attenua- 


tion characteristic having zero attenuation 
below w = 1 and a net gain of tt nepers there- 
after. In other words, the amplitudes of all 
frequencies below o> = 1 are increased by a fac- 
tor of about 22 to 1. If the region of added 
gain is pushed to a higher frequency or con- 
centrated within a narrow band, the multi- 
plying factor rapidly becomes larger. For ex- 
ample, if we maintain A at approximately zero 
below w = 2, the average gain above this point 
must be 27r nepers, corresponding to a multi- 
plying factor of 500 to 1. We secure the same 
factor by attempting to concentrate the region 
of negative attenuation in the band between 
o> = 1 and 0 ) = 2. The multiplying factor also 
goes up rapidly as we increase the prediction 
time. For example, with the gain uniformly 
spread over the frequency region above w = 1 
the multiplying factor is 500 for a prediction 
time of 4 seconds, or more than 10,000 for a 
prediction time of 6 seconds. 

Reasonable multiplying factors with long 
prediction times can be obtained only by carry- 
ing the negative attenuation region to very low 
frequencies. As indicated previously, the cost 
of this is an increase in the time required for 
the signal to change from one constant or 
nearly constant value to another. For exam- 
ple, in the first illustration above, if the region 
of TT nepers net gain is carried down from 
oj = 1 to w = 0.2 the integral in (3) is just five 
times as great as it was before, so that the 
characteristic corresponds to a prediction time 
of 10 rather than 2 seconds. This change 
would correspond to an increase^' from perhaps 
4 or 5 to perhaps 20 or 25 seconds in the time 
required for the circuit to settle from one con- 
stant value to another. 

Practical examples of the transmission char- 
acteristics of overall prediction circuits, with 
particular emphasis on the dominant effect of 
even very small negative attenuations at ex- 
tremely low frequencies, are shown later in 
Figures 5 to 8, inclusive. In the linear predic- 
tor, A — Ao varies as — nears zero, and it is 
easily seen that such a term makes a finite con- 


^ Only rough numbers can be given, since circuits 
with the square-cornered attenuation characteristics 
chosen for illustrative purposes would have very ripply 
transient characteristics, corresponding to no very well 
marked settling time. 


iJggafTdFVTIa^ 


OVERALL CHARACTERISTICS OF PREDICTING NETWORKS 


99 


tribution to the integral in (3) . On the other 
hand, the attenuation of the quadratic predic- 
tor, which is capable of dealing exactly with 
polynomial functions of time of the second 
degree or less, is necessarily zero at the origin^ 

^ Ci the discussion of Quasi-Distortionless Prediction 
Networks in Appendix A. 


to terms of the order of m *, so that the integral 
in this region can be neglected. This slight 
difference between the two characteristics at 
frequencies of the order of 0.01 cycle per 
second and below is sufficient to balance the 
obviously greater negative attenuation of the 
quadratic predictor at higher frequencies. 


# /CONFIDENTIAL 


Chapter 9 


THE ASSUMPTION OF ANALYTIC ARCS 


T he discussion in the previous two chap- 
ters has been based upon the assumption 
that the least squares criterion forms a suita- 
ble measure of performance for a predicting 
network. This assumption permitted us to re- 
strict our attention to the amplitude spectra 
of the signal and noise, leaving phase relations 
entirely out of account. Thus, both signal and 
noise could be thought of as “random noise” 
functions characterized by random phases and 
Gaussian distributions, as described in the 
preceding chapter. So far as the noise is con- 
cerned, there seems to be nothing wrong with 
this assumption. In the case of the signal, how- 
ever, it appears that significant phase relations 
may exist. This chapter will consequently set 
up an alternative analysis which permits the 
significance of possible phase relations in the 
target paths to be estimated. 

The alternative analysis is based upon the 
assumption that the target courses are sequen- 
ces of analytic segments of different lengths 
joined together. These segments are simple 
predictable curves such as straight lines, pa- 
rabolas, and circles. Significant phase relations 
are implied by the assumption that there are 
sudden changes from one type of course to 
another. 

This picture of target paths is, of course, 
extreme. There are no such sharp discontinui- 
ties between one segment and another, nor do 
airplanes fly perfectly along simple curves 
even for limited periods. Nevertheless, it is 
the conception of target courses upon which 
the rest of our analysis is based. The reasons 
for believing that it is a closer approximation 
to actual target courses than, say, a random 
noise function with the same power spectrum 
would be, are given later. Perhaps more im- 
portant is the fact that the possibility of hit- 
ting an airplane flying along such a simple 
analytic arc is much greater than it would be 
if we were attempting to predict a correspond- 
ing random noise function. It is thus advan- 
tageous to take the analytic arc assumption as 
a basis for designing the prediction circuit. 


even if the assumption seems to be reasonably 
well justified over only occasional segments of 
actual target paths. An example of such a 
situation is furnished by the bombing run 
illustration described in Chapter 7. 

As a corailary to the analytic arc assump- 
tion it is also assumed that the theoretical 
predicted point must be quite close to the actual 
target position if the probability of scoring a 
hit is to be appreciable. In other words, such 
dispersive factors as random errors in com- 
puter or gun or the lethal radius of the shell, 
which would tend to produce occasional hits at 
long distances from the theoretical predicted 
point, are quite small. This is such a plausible 
assumption in the light of present-day antiair- 
craft experience that its critical importance in 
the present argument is likely to go unper- 
ceived. However, this is the assumption which 
limits consideration to small errors in predic- 
tion, whereas the least squares criterion natu- 
rally gives greatest emphasis to large errors. 
If, for example, antiaircraft projectiles were 
suddenly endowed with a much greater de- 
structive radius, we would be much more in- 
terested in fairly large misses, and the objec- 
tions to the least squares criterion would disap- 
pear. 

These postulates are discussed in more detail 
in the following sections. In anticipation of 
this discussion the following conclusions may 
be mentioned: 

1. With the assumptions as stated, the pre- 
diction should be on a modal rather than a 
least squares basis. In other words, the gun 
should be aimed at the most probable future 
position of the target. 

2. Modal prediction requires evaluation of 
the parameters of the analytic arc the target 
is at present traversing. This can be accom- 
plished by smoothing the values of these pa- 
rameters evaluated for a period in the past. 

3. If the smoothing is performed by linear 
invariable networks, the impulsive admittances 
of these networks should have a definite cutoff 
after a finite smoothing time. By this means 




:q>fid£Mial 


3 


100 


THE POISSON DISTRIBUTION OF SEGMENT END POINTS 


101 


all data over a certain age are given zero weight. 
The method of calculating the proper smooth- 
ing time is developed. 

4. Definite advantages can be obtained from 
circuits with variable smoothing times if such 
systems can be satisfactorily mechanized. 

9 1 THE TARGET COURSES 

The target courses, like the tracking errors, 
can be thought of as a statistically generated 
set of functions — that is, a stochastic process. 
The structure of this process is, however, very 
different from that of the tracking errors. It 
is by no means satisfactory to assume the 
target courses to be equivalent to a random 
noise having the same power spectrum as the 
target courses. As we pointed out in Chapter 
7, the target is piloted by a purposeful human 
being. It tends to follow a definite simple curve 
for a period of time and then to shift to a new 
simple curve. Much of the flight is in attempted 
straight lines with constant velocity. Most of 
the remainder can be considered to be segments 
of circles or helices in space, or as segments of 
parabolas or higher degree curves. Straight 
line constant speed flight corresponds to the 
airplane controls in a neutral position. The 
helical flight is a natural generalization allow- 
ing arbitrary, but fixed, positions of the con- 
trols. The curves which are parabolic functions 
of time correspond to constant acceleration in 
the three space coordinates. Thus, all these 
assumptions have a reasonable physical back- 
ground. 

Most antiaircraft computers are constructed 
on the assumption of straight line flight, al- 
though some work has been done in World 
War II on curved flight directors both with the 
helical and the parabolic assumptions. There is 
not a great deal of difference in these two 
generalizations from the practical point of 
view, since determination of acceleration terms 
is subject to such large errors in any case. 

The important part of this representation 
of the target courses is that they consist of 
segments of simple analytic curves joined to- 
gether. The individual segments are completely 
predictable if we have a part of the segment 
given exactly. One need merely evaluate the 
parameters of the segment from the given part 


and evaluate the curve for t = tf. The unpre- 
dictable part of the target courses is due to the 
possibility of sudden changes from one segment 
to another. With random noise functions the 
unpredictableness occurs continuously. 

This simplified description of the target 
courses as piecewise analytic functions must 
be recognized as only a first approximation. A 
more complete description of the target course 
would include the “fine structure,” the con- 
necting curves between the various analytic 
segments and the deviations from the segments 
due to random air disturbances and similar 
causes. This latter effect, the wandering of the 
target from its intended path, might be reason- 
ably well represented by the addition of a 
random noise function to the piecewise analytic 
functions described above. 

9 2 the POISSON DISTRIBUTION OF 
SEGMENT END POINTS 

The analytic segments of which the course 
is supposed to consist are not all of the same 
duration — we may assume some probability 
distribution of the duration of these segments. 
The simplest assumption here is that the 
breaks occur in a Poisson distribution in time. 
This assumption is not necessary for our 
analysis but is a reasonable one and leads to 
a simple mathematical treatment. Any other 
reasonable distribution would give comparable 
results. 

A series of events is said to occur in a 
Poisson distribution in time if the periods be- 
tween successive events are independent in the 
probability sense and are controlled by a distri- 
bution function 

p{l)dl = i dl . 

Here p {l)dl is the probability of an interval of 
length between I and I + dl. This means that 
the frequency of intervals of a given length is 
a decreasing exponential function of the length. 
This type of distribution is familiar in physics 
as describing the decay of radioactive sub- 
stances. The time a in the distribution function 
is the average length of the intervals, since 

r= jT I pii)di 


EONFIDEXTIA; 


102 


THE ASSUMPTION OF ANALYTIC ARCS 


00 

- e-'/- dl 
a 

= a . 

It is related to the ‘‘half life’' h of the interval 
by 

6 = a In 2 . 

The single number a completely specifies the 
Poisson distribution. The events may be said 
to be happening as randomly as possible apart 
from the fact that they occur at an average 
rate of 1/a per second. 

Another way of describing a Poisson distri- 
bution of events is the following. The probabil- 
ity of an event in a small interval of duration 
dl is (l/a)dl and is independent of whether or 
not events have occurred in any other nonover- 
lapping intervals. 

93 THE PROBABILITY DISTRIBUTION 
OF FUTURE POSITIONS 

Let us suppose that we have a record of the 
course of the target up to the present time and 
a complete statistical description of the set of 
target courses. What can then be said about the 
position of the target tf seconds from now? If 
we were able to analyze the data completely 
the most we could obtain would be a probability 
distribution function for the future position. 
This distribution function would give the prob- 
ability, in the light of the course history, of 
the target being at any point in space at the 
future time. This function would assume large 
values at likely points and low values at un- 
likely points. For tf small the distribution 
would be highly concentrated and for larger tf 
it would tend to spread out. 

In the simple case we have been discussing, 
of a Poisson distribution of sudden changes in 
type of course, the distribution consists of two 
parts. First, there is a spike of probability at 
one point, the continuation of the present pre- 
dictable segment. Second, there is a continuous 
distribution which corresponds to possible 
changes to a new segment during the time of 
flight. As tf increases the total probability in 
the spike decreases exponentially toward zero, 
and the total in the continuous part increases 
exponentially toward unity. The behavior is 
roughly as indicated in Figure 1. 



Figure 1. Probability distribution of future po- 
sition of target, assuming piecewise analytic 
courses. 

A very different type of future position dis- 
tribution is exhibited with other assumptions 
about the target courses. For example, suppose 
the courses were random noise functions with 
the power spectrum 


A typical noise function with this spectrum is 
shown in Figure 2. In Figure 3 is shown a 
typical velocity under the other assumption, 
that the courses are piecewise analytic and in 
fact straight lines between breaks. If the 
breaks are Poisson distributed, both Figure 2 
and Figure 3 have the same power spectrum, 
l/(a2 + tu^). The future distribution of veloci- 
ties for Figure 3 is shown in Figure 1, and for 
Figure 2, it will be as shown in Figure 4. In the 
random noise case the future distribution is a 



CONFIDENTIAL^ 



THE PROBABILITY DISTRIBUTION OF FUTURE POSITIONS 


103 


Gaussian distribution with no spike. The center 
of this distribution decreases exponentially to- 
ward zero with increasing time of flight ac- 
cording to the formula 

where is the present value of the function 
and Xtf is the mean of the future distribution. 



Figure 2. Typical noise function. 


The standard deviation <t of the distribution in- 
creases exponentially toward the rms value of 
the function according to 

(7 = A(1 - 

Supposing that this distribution function 
could be determined, where should the gun be 
aimed? The answer to this will depend on two 
factors: the gun dispersion, and the lethal 


t 


Figure 3. Typical velocity function. 


effects of the shell. If the gun is aimed to 
explode the shell at a certain point in space, 
the shell will not necessarily explode at that 
point, but rather there will be a distribution of 
positions centered about the point aimed at, 
because of gun dispersion. Also, if the shell 
explodes at a certain point and the target is at 


another point, there will be a certain proba- 
bility of lethal effect which decreases rapidly 
with increasing distance between the points. 
These two functions could be combined by a 
product integration to give the probability of 
lethal effect if the target is at one point and 



Figure 4. Probability distribution of future posi- 
tion of target, assuming courses with random 
noise properties. 


the gun aimed to explode the shell at a second 
point. To determine the probability of a hit 
when aiming at a certain point, then, we should 
multiply the probability of the target being at 
each point in space by the probability of lethal 
effect when it is at that point and integrate the 
product over all space. The optimum point of 
aim will be the one which maximizes this in- 
tegrated product. 

In one dimension this may be expressed 
mathematically as follows. Let Pix) be the 


XFIDENTIAL 




104 


THE ASSUMPTION OF ANALYTIC ARCS 


future position distribution of the target, so 
that P(x)dx is the probability of it being in 
the interval from xtox dx sX the future time. 
Let Q{x,y) be the probability of hitting the 
target if the gun is aimed at point ij and the 
target is at point x. Then the total probability 
of a hit when aiming at point y is 

R{y) = J P{x) Q{x,y) dx . 

The point of aim y should be chosen to maxi- 
mize R{y). 

In the cases we consider, the lethal radius of 
the shell and the dispersion of the gun are both 
assumed to be small in comparison with the 
range of future positions if there is a change 
of course during the time of flight. This means 
that Qix,y) is small unless x is xery near to y. 
Q{x,y) can be, in fact, considered to be a 8 
function of (x-y), and the value R(y) is then 
just a constant times P(y). Thus, the best 
aiming point under this assumption is the most 
probable future position of the target. The as- 
sumption of small lethal distance is generally 
valid with antiaircraft fire and ordinary chemi- 
cal explosive shells. 

Now the most probable future position in our 
case is the spike of probability corresponding 
to the analytic extrapolation of the present seg- 
ment of the target course. To determine its 
position one must find the parameters of this 
segment and evaluate for tf seconds in the 
future. For example, if the segments are as- 
sumed to be straight lines (constant velocity 
target) the velocity components are determined 
and multiplied by tf to give the predicted 
change in position. These changes are added to 
the present position to give the future position. 
If helical or parabolic segments are assumed, 
the parameters of these curves are determined 
from the past data, and the curves extrapo- 
lated tf seconds into the future. 

These conclusions may be contrasted with 
the idea of aiming at the point which mini- 
mizes the mean square error. The least squares 
criterion amounts to aiming at the mean or 
center of gravity of the future distribution of 
position. This point will ordinarily be under 
the continuous part of the distribution and not 
at the spike; e.g., the point marked in Figure 1. 
Its position depends to a considerable extent on 


distant parts of the distribution, which would 
surely be complete misses in any case. The 
chief advantage of the least squares criterion 
is that it fits in well with the mathematical 
tools suitable to these problems, leading to 
solvable equations. 

The least squares criterion will still appear 
in our analysis in that we attempt to smooth 
our course parameters in such a way as to 
minimize the mean square error in these, a 
very different thing from minimizing the mean 
square error in the predicted position of the 
target. 

94 NECESSITY OF A SHARP CUTOFF 

The changes in the course parameters be- 
tween adjacent segments can be very large. 
Also, at the start of operations and in changing 
from one target to another there will be large 
and erratic variation of the input to the 
smoothing and predicting circuits, unrelated to 
the present target course. If any of these data 
are used in prediction, the result will almost 
surely be a miss because of the small lethal 
radius of the shell. The only way to eliminate 
these errors in a linear invariable system is to 
have all weighting functions cut off sharply 
after a short time. Then all data over a certain 
age are eliminated. Hits will occur only when 
the target has been on a predictable segment for 
this length of time or more and remains there 
at least tf seconds in the future. 

Suppose the weighting function for velocity 
has a 1 per cent tail beyond the cutoff point 
and that the trackers start following the target 
from a zero position. Then after the smoothing 
time there will be, because of the lack of exact 
cutoff, a 1 per cent error in velocity. If the 
time of flight were 15 seconds and the target 
velocity 200 yards per second, this represents 
an error of 30 yards in predicted position. 
Since this is comparable to the other errors in 
a typical director, we conclude that the tail of 
the smoothing curve should not be much greater 
than 1 per cent of its total area. 

95 CALCULATION OF THE BEST 

SMOOTHING TIME 

Under the assumptions we have made, the 
proper smoothing time to maximize the number 
of hits can be determined as follows. Let P(l) 


I^CONFIDENTI^ 


CALCULATION OF THE BEST SMOOTHING TIME 


105 


be the probability that a predictable segment 
of the course lasts for I seconds or more. In 
the Poisson case this function is 

P(/) = . 

With a given smoothing time there will be a 
certain probability of hitting the target, as- 
suming it has been on the present segment for 
5^ seconds in the past and will remain there for 
tf seconds in the future. We assume changes 
in course to be so large that any change re- 
sults in a miss. This probability of a hit Q {S ) , 
provided it remains on the course, will be an 
increasing function of S. Ordinarily the stand- 
ard deviation will decrease as the square root 
of the smoothing time. We have assumed the 
lethal radius of the shell small compared to the 
dispersion of shells about the target. The prob- 
ability of a hit will then vary inversely with 
the volume through which the shells are dis- 
persed. If the gun itself had no dispersion but 
all errors were due to tracking errors (and if 
the tracking error spectrum is flat), the prob- 
ability of a hit would then vary as for 

S in the region of interest. This is because 
there are three dimensions and the expected 
error in each of these is decreasing as 5“^/^ 
With gun dispersion present, Q(S) will have 
the form 

Q(S) = K{a\ + al^ 

where is the standar d de viation due to the 
gun dispersion, and o-sVa/'S' that due to track- 
ing errors. The sum of the squares is the total 
variance in each dimension and the three- 
halves power gives the total dispersion volume. 

When these two functions P{1) and Q(S) 
are known, the best smoothing time is that 
which minimizes the product 

P{S + t) • Q{S) . 

The first term is the probability of a predict- 
able segment of the course lasting S tf sec- 
onds, and the second term is the probability of 
a hit if it does last that long. Therefore, the 
product is the probability of a hit with smooth- 
ing time S. 

In the Poisson case, with no gun dispersion, 
the calculation is as follows : 

P{1) = e 


« + </ 

P{S + tj) = e ^ 

Q{S) = G 

f{S) = P(S + lf)Q(S) = Be-s/- 53/2 

/'(S) 0 

5 = |a 

The proper smoothing time is % of the aver- 
age segment length, and is independent of the 
time of flight and all other factors. 

The presence of gun dispersion and computer 
errors which are independent of smoothing 
time decreases the best S from this value. In 
this case the equation for optimal S is the 
quadratic 

J l<^2- — 0^2 = 0, 

\a / a 2 

hence 

S — 0-2 + + b 0-2 

a ~ 2(7? 


Here (Tj is the part of the errors which is in- 
dependent of smoothing time (dispersion 
errors in the computer, etc.) and (73 is the error 
which varies inversely with the square root of 
Sy ( 7 i being its value at S = a. Ordinarily is 
several times 0-2 in which case we have approxi- 
mately 

S ^02 

a (7i \ 2 

There are other factors which we have neg- 
lected, which decrease the best smoothing time 
still further. The wandering of the target about 
the predictable segments assumed in the above 
simplified analysis makes old data less reliable 
and therefore reduces S. Also, there is the tac- 
tical consideration that when starting to track 
a target it is desirable to commence firing as 
soon as possible, even if reducing this time 
makes individual hits somewhat less probable. 
For these and other reasons the best smooth- 
ing time will be just a fraction of a. 


(yi\/(y\ + b (7^ 


2(7? 




\CONFIDENTIA 


106 


THE ASSUMPTION OF ANALYTIC ARCS 


NONLINEAR AND VARIABLE 
SYSTEMS 

The compromise required in choosing a cer- 
tain definite smoothing time can be eliminated 
by the use of nonlinear elements. In particular, 
if a method is devised for determining when 
changes of course occur, this indication can be 
used to start a new linear but variable smooth- 
ing operation, so that the device uses all the 
data pertinent to the present segment and no 
data from previous segments. There is a clear 
improvement in such cases although not so 
great as might be expected. There are many 
practical difficulties in proper adjustment of 
such a '‘trigger” action. If the trigger is too 
sensitive it will assume new segments due 
merely to tracking noise and seldom allow suffi- 
cient smoothing for accurate fire. If it is too 
insensitive it fails in its function of quickly 


locating changes of segment. Since the noise 
and target courses are subject to considerable 
variation, this adjustment is not easy. 

In such a system the smoothing may be 
linear — the only nonlinearity is the tripping 
circuit. The analysis of best weighting func- 
tions, etc., given in later chapters can for the 
most part be applied to such cases. There may 
also be advantages to be derived from making 
the smoothing operator depend on the general 
position in space of the target relative to the 
gun. The smoothing time may be varied, for 
example, as a function of the time of flight. 
This type of variation would be slow compared 
to the noise frequency, and here again the 
linear analysis can be used. 

Whether any real advantage can be obtained 
by “strongly” nonlinear smoothing in practical 
cases other than these two possibilities is ques- 
tionable. 


CONFlDKVn \i. 


1 


Chapter 10 

SMOOTHING FUNCTIONS FOR CONSTANTS 


T he analytic arc assumption described in 
the previous chapter immediately allows us 
to reduce a vast proportion of data-smoothing 
problems to a relatively concrete form. Obvi- 
ously the arc will be specified by a number of 
parameters and the principal object of the com- 
puting and data-smoothing circuits must be to 
isolate values of these parameters on the basis 
of which a prediction can be made. In practi- 
cal cases the instantaneous values of the 
parameters are isolated by coordinate con- 
verters. The function of the data-smoothing 
circuit is to provide a suitable average from 
these instantaneous values. This is called 
“smoothing a constant” here since the param- 
eters are assumed to be constant along each 
arc, although they may change radically from 
one arc to another. 

The data-smoothing network is most con- 
veniently specified by its impulsive admittance. 
(See Appendix A.) In accordance with the 
assumptions made in the previous chapter, it 
will be assumed that the desired impulsive ad- 
mittance is identically zero after some limiting 
time T. Thus, T seconds after a change from 
one analytic arc to the next the new parameter 
value is established. T is the so-called “settling 
time” of the data-smoothing network. 

With the settling time limit given, the prob- 
lem of choosing a suitable data-smoothing net- 
work reduces to that of finding the best shape 
of the impulsive admittance characteristic for 
t <T. Obviously this shape determines how 
the output of the network changes in going 
from the parameter value appropriate for the 
first arc to that appropriate for the second. The 
exact way in which the response settles from 
one constant value to the next is, however, 
usually of comparatively little interest. The 
shape of the weighting function is of impor- 
tance chiefly because of its effect on the noise. 
For each noise spectrum there is, in principle, 
an optimum shape for the weighting function. 
The present chapter approaches the problem of 
choosing a shape which will minimize the effect 
of noise from several points of view. 


It should be noted that the term noise as used 
here does not necessarily refer to the errors 
associated directly with the tracking data. The 
tracking data may have been subjected to co- 
ordinate conversions, differentiations, or other 
processes of computation before reaching the 
data-smoothing network."" The noise associated 
with the signal to be smoothed thus will usually 
have characteristics differing from those of the 
noise associated with the tracking data. 

1 EXPONENTIAL SMOOTHING 

Before attacking the problem of smoothing a 
constant in a systematic way it is worth while 
to consider an important special case. This is 
the so-called exponential smoothing circuit. It 
leads to a data-smoothing network in which 
the output V is related to the input E by 

/ CD 

E{t — r) e~^'^ dr 

SO that the impulsive admittance W (t) is an 
exponential function of time, as illustrated by 
Figure 1. 



Figure 1. Simple exponential weighting function. 

An impulsive admittance of the type shown 
in Figure 1 does not show any very definite 
settling time. The exponential curve ap- 
proaches zero gradually, and it is a long time 
after a change in course before the effects of 
the data obtained on the old course are negli- 
gible. This is obviously an undesirable result. 


^ In exceptional circumstances the physical apparatus 
in which these processes are carried out may also be 
sources of additional noise. 



107 



108 


SMOOTHING FUNCTIONS FOR CONSTANTS 


and the exponential weighting function is con- 
sequently not a recommended one for situations 
to which the analytic arc assumption applies. 
The exponential solution is, however, described 
here because it occurs in such a vast variety of 
cases. It is found, in fact, whenever the data- 
smoothing device is specified by a linear first- 
order differential equation with constant coeffi- 
cients. It may thus correspond to many simple 
situations. For example, this is the result 
which would be obtained in an electrical circuit 
if we smoothed the data by placing a simple 
shunt capacity across a resistance circuit. In 
mechanical structures it is encountered when- 
ever the damping depends either upon simple 
inertia or a simple compliance. 

Simple exponential smoothing also occurs in 
a variety of other situations which may be 
somewhat less obvious. For example, it is the 
effective result in either an aided laying or a 
regenerative tracking scheme whenever the 
ratio between rate and displacement correc- 
tions is fixed. Another somewhat similar ex- 
ample is furnished by the feedback amplifier 
circuit shown in Figure 2. Since rapid fluctua- 



Figure 2, Feedback amplifier circuit giving simple 
exponential weighting function. 


tion give exponential smoothing, which has no 
very well-marked settling time, it is clear that 
a data-smoothing network which shows a well- 
defined settling time must probably be at least 
moderately complicated. 

2 CURVE-FITTING METHOD 

Consider the signal E shown in Figure 3 
under the assumption that the true signal is 
constant and the superposed noise is random 



Figure 3. Piecewise constant signal with noise. 

with a flat spectrum. The best constant A, in 
the least squares sense, which can be fitted to 
the signal from ^ — T to ^ is that which mini- 
mizes 

f [A - F(X)]2 d\ , 

Jt-T 

viz., 

A = E{\) rfX . (1) 

Comparing this with equation (2), Appendix 
A, it will be seen that A, which is obviously a 
function of t, is the response to the assumed 
signal of a network whose impulsive admit- 
tance is 


tions in the output of this amplifier are fed 
back through the capacity and tend to oppose 
the input voltage, the structure acts as a 
smoother, and more detailed analysis would 
show that it has characteristics similar to those 
obtained by using a shunt capacity across a 
resistance circuit. The structure is introduced 
here because considerable use is made of it in 
connection with the discussion of nonlinear 
smoothing in a later chapter. 

One simple conclusion about data-smoothing 
networks can be drawn immediately from this 
discussion. Since all structures simple enough 
to be specified by a first-order differential equa- 


W(0 = ^ 0 <f <T. (2) 

This is the best weighting function for smooth- 
ing under the assumed circumstances. It is 
illustrated in Figure 4. 

A more complex situation is one in which the 
true signal is a line of constant slope with 



Figure 4. Best weighting function for smoothing 
piecewise constant signal. 


OXFIDKM lAll 


AUTOCORRELATION METHOD 


109 


superposed flat random noise, as shown in Fig- 
ure 5. For convenience the analysis will be 
conducted in terms of the age variable r = t — k. 



B 


- r 

T^o 


E'it - r) -riT - t) dr. 


W{t) 


= I • |(i - l) 0 < < < r. 


It should be noted also that the right-hand 
member of the first of equations (3) is form- 
ally the same as that of equation (1). Hence 
the response of the network specified by (2) 



Figure 5. Piecewise linearly varying signal with 
noise. 

The best straight line A — Bt which can be fit- 
ted to the signal from t = 0 to t = T is that 
which minimizes 

fj [A - Bt- E{t - T)]^dr. 

Hence A and B must satisfy simultaneously 

A-\b= ^JjE{t-r)dr 

Eliminating A, we get 

whence by partial integration 


(3) 


Figure 6. Best weighting function for smoothing 
piecewise linearly varying signal. 

and illustrated in Figure 4, to the type of 
signal shown in Figure 5, will correspond to 
the value on the best straight line T/2 seconds 
back from t, the present time. This network is 
still the best for smoothing the signal, but it 
introduces a delay of one half of the smooth- 
ing time. The delay may be reduced only at 
the price of a reduction in smoothing unless the 
smoothing time is increased. 


103 AUTOCORRELATION METHOD 

The autocorrelation method with finite set- 
tling time was first used by G. R. Stibitz in 
numerical determination of the best weighting 
function for smoothing the derivative of track- 
ing data with typical tracking errors. This 
method was also used to determine the sensitiv- 
ity of smoothing to departures of the weighting 
function from the best form. 

The analysis is based upon the formula 


V(t) = 




g'(t - r) • Wir) dr t> T 


Comparing this with (7), Appendix A, it will 
be seen that B, which is obviously a function of 
t, is the response to the derivative of the as- 
sumed signal of a network whose impulsive 
admittance is 


(4) 


for the response to the derivative of the error 
time function g{t) of a network whose impul- 
sive admittance or weighting function W {t) is 
identically zero for t > T as well as for ^ < 0. 
Since measured tracking errors are generally 
tabulated only at I-second intervals, the in- 
tegral may be approximated by the sum 


This is the best weighting function for smooth- 
ing the derivative of the signal under the as- 
sumed circumstances. It is illustrated in Fig- 
ure 6 and is generally referred to as the “para- 
bolic weighting function.” 


V{t) = I 


• w 

m+C>i) m-iH) 


for integral values of t. 

The instantaneous transmitted power is the 


NFIDENTIA 


110 


SMOOTHING FUNCTIONS FOR CONSTANTS 


square of this expression, and the average 
transmitted power is 

N 

P,v8 = lim J. V yuA . 

N ^ 

This may be expressed in the form 

T 

^avg = EH ^ Mx/C' 'W (5) 

m,n = 1 

where 

N 

^ lim 1 y< A .A 

^m-n Ar_>oo Lk 
(=0 

is the autocorrelation of the errors. Having 
computed the autocorrelation, (5) may be mini- 
mized with respect to the IT’s by familiar 
methods, under the constraint 

i 1 • 

m = 1 

The values of IT thus obtained are the speci- 
fication of the best weighting function.^ Equa- 
tion (5) may then be used to determine the 
sensitivity of smoothing to departures of the 
weighting function from the best form. 

Proceeding along this line, Stibitz found that 
the best weighting function for typical actual 
tracking errors was generally intermediate to 
the uniform and parabolic ones shown in Fig- 
ures 4 and 6. Furthermore, Stibitz found 
that the difference in smoothing obtained from 
the best weighting function on the one hand 
and from the uniform or the parabolic weight- 
ing function on the other hand, is negligible in 
practice. 

The autocorrelation method was later for- 
malized by R. S. Phillips and P. R. Weiss who 
incorporated it into a theory of prediction.'^ A 
brief exposition of this formulation is given 
in Appendix B. 

^ ELEMENTARY PULSE METHOD 

For the purposes of this method, an ele- 
mentary noise pulse is defined by a time func- 
tion Fo(t) which satisfies the following require- 
ments : 

1. Identically zero when ^ < 0. 

^ The computations involved may be considerably re- 
duced by noting the symmetry property proved in Sec- 
tion B.2, Appendix B. 


2. Contains no terms which increase expo- 
nentially with time. 

3. Power spectrum N (oj^) is the same as that 
of the noise. 

The noise is then regarded as the result of 
elementary noise pulses started at random. 
Alternatively, it may be regarded as the result 
of flat random noise passed through a network 
whose transmission function is S(p) = L 
[Foit)]. As a matter of fact, only S{p) is 
required in the analysis, and this is readily de- 
termined from the relation 

|,SM|2 = N(cc^) , 

together with the condition that S{uo) cor- 
responds to the transmission function of a 
minimum-phase physical structure (cf. Appen- 
dix B) . 

The response F (t) to the elementary noise 
pulse F^it) of a network whose impulsive ad- 
mittance is W (t) is given by the operational 
equation 

F{t) = S{p) • Wit) 

in accordance with the footnote in Section A.5, 
Appendix A. The best form for IT (^) is there- 
fore that which minimizes the integral 



[Fit)? dt 


under the restriction 


( 6 ) 


when to > T . 



Wit) dt = 1 


(7) 


This is as much of the elementary pulse 
method as we shall need in order to reconsider 
the cases treated in Section 10.2. For the treat- 
ment of more general cases the method is de- 
scribed in greater detail in Appendix B. 

The minimization of the integral (6) under 
the restriction (7) reduces to a simple isoperi- 
metric problem in the calculus of variations, in 
cases in which S(p) is a polynomial in p. It is 
essential first of all, however, to note that if 
S(p) is of degree n, the integral (6) will con- 
verge only if IT(^) is differentiable at least n 
times. In other words, IT(^) must have con- 
tinuous derivatives of all orders up to the 
(^— l)th inclusive, although the ni\i derivative 
may have finite discontinuities. In particular, 
if W {t) is to be zero outside of 0 < ^ < T, its 



ELEMENTARY PULSE METHOD 


111 


derivatives of orders up to the (n— l)th inclu- 
sive must vanish at both t = 0 and t = T. These 
2n boundary conditions must be imposed on the 
solution of the Euler equation which in this 
case is 

A is a constant parameter which is finally ad- 
justed to that the restriction (7) is satisfied. 

The first case treated in Section 10.2 is one 
in which AT (w^) = 1, whence 5 (p) =landF(0 
= W(t). The integral (6) is a minimum under 
the restriction (7) if W (t) is constant by 
intervals. The restriction (7) then requires 
W (t) to be of the form (2) . 

The case of first derivative smoothing treated 
in 10.2 is one in which N (w^) = whence S (p) 
= p and F(t) = W {t) . If the integral (6) is to 
converge at all, W {t) must not have discon- 
tinuities of impulsive or higher type; in other 
words, W (t) must be continuous through all 
values of t. The integral is a minimum under 
the restriction (7) if W {t) is constant by 
intervals. The restriction (7) then requires 
W (f) to be of the form (4) . 

These results may be generalized immedi- 
ately. In whatever way the signal to be 
smoothed may have been derived from the 
tracking data, let the power spectrum of the 
noise associated with it he N {(o^) = Then 
S(p) =p”andF(f) = (f) • If the integral 


(6) is to converge at all, (t) must be con- 

tinuous through all values of t. The integral is 
a minimum under the restriction (7) if 
(t) is constant by intervals. The restric- 
tion (7) then requires W (t) to be of the form 

It may be noted that the convergence re- 
quirements which arise in the foregoing dis- 
cussion are directly related to the discussion 
and theorem in Section A.8, Appendix A, with 
respect to the relationship between discontinui- 
ties in the impulsive admittance and its deriva- 
tives on the one hand, and the ultimate cutoff 
characteristic of the transmission function on 
the other hand. The continuity of (t) is 

obviously required to make the transmission 
fall off ultimately at the rate of 6(n + l) db per 
octave against the rise of 6n db per octave in 
the noise power spectrum. 

The integral (6) may also be used to evalu- 
ate the relative advantage of the best weighting 
function over another weighting function. As 
an example, consider the case where the weight- 
ing function (2) is the best. The value of the 
integral (6) in this case is 1/T. If the weight- 
ing function (4) is used against the same noise, 
the value of the integral (6) is 6/5T. Hence, 
as far as rms error or standard deviation is 
concerned, the second weighting function is 
V5/6 or 0.913 as efficient as the first. 



Chapter 11 

SMOOTHING FUNCTIONS FOR GENERAL POLYNOMIAL EXPANSIONS 


T he theory of “smoothing a constant” de- 
veloped in the preceding chapter will be 
extended in this chapter to the problem of 
smoothing a polynomial function of time of any 
prescribed degree. The extension is, however, 
restricted to the case of a flat noise spectrum. 
In addition to the smoothing problem, the 
analysis also provides a way of designing a 
network which will extrapolate the polynomial 
a given distance tf into the future. The network 
is so arranged that tf is continuously variable. 
In addition, the degree of the polynomial can 
readily be changed to fit changes in the com- 
plexity of the assumed form of the data, apart 
from noise. 

It is clear that these results amount, in a 
certain sense, to an alternative to Wiener’s 
method for the design of prediction circuits for 
general time series. Thus, to predict a time 
series of any given complexity we would need 
only to begin with a polynomial of sufficiently 
high degree to fit the observed data, and extra- 
polate. Aside from the restriction to a flat 
noise spectrum, perhaps the most obvious dif- 
ference from Wiener’s method is the fact that 
the settling time restriction limits the data 
upon which the prediction rests to a finite in- 
terval in the past. To advance such a prediction 
theory seriously, however, it would be neces- 
sary to go much farther into the way in which 
the degree of the polynomial is established and 
the justification for assuming that the extra- 
polated value represents a probable future 
value for the function.^ 

This general discussion will not be under- 
taken here. Since prediction with high degree 
polynomials will certainly be sensitive to minor 
irregularities in the data, tracking errors 
would necessarily limit the application of the 
method in any case. If we confine ourselves to 
reasonably low degree polynomials, however, 

^ As an example of possible difficulties we may notice 
the fact that two polynomials of different degree which 
approximate a given function as closely as possible, in 
a least squares sense, in a prescribed interval fre- 
quently differ radically outside that interval. 


112 


the method is useful. An example is furnished 
by the prediction of airplane position, in rec- 
tangular coordinates, by quadratic functions of 
time. Here the square terms represent the 
effects of accelerations in the various coordi- 
nates. We can defend the inclusion of such 
terms on the ground that it is plausible to as- 
sume that an airplane may experience constant 
accelerations, due to turns, the force of gravity, 
etc., for considerable periods of time. The 
linear term represents plane velocity and needs 
no defense. The constant term, of course, gives 
the plane position at some reference time. In- 
cluding it in the smoothing operation is equiva- 
lent to introducing “present-position” smooth- 
ing of the sort suggested by the broken lines 
in Figure 1 of Chapter 7.*" 

Aside from its direct interest as a possible 
prediction method, the analysis in this chapter 
is also of indirect interest for the additional 
light it sheds on the effect of the noise spec- 
trum on smoothing functions. It turns out that 
smoothing a power of time, with a flat noise 
spectrum, is equivalent to smoothing a constant 
with a somewhat different noise spectrum. 
Thus the smoothing functions developed for 
polynomials are also useful as special cases of 
smoothing functions applicable to constants. 

1 GENERAL METHOD 

Let X be any past value of time and let t be 
the present value. If the data is fitted with a 
smooth curve E (A) , the predicted value may be 
taken as E{t + tf) . The procedure of fitting is 
the familiar one of minimizing the integral 

t 

[ E(\) - E{\) f Wo(f,X) d\ 


In the circuit of Figure 1, Chapter 7, however, the 
smoothing network would produce a lag in the present- 
position data delivered to the prediction circuit, and 
this lag would, of course, mean some error in follow- 
ing a moving target. In the method described in this 
chapter such lags are automatically compensated for 
by adjustments in the coefficients of the other terms of 
the polynomial. 




confidentia; 





GENERAL METHODS 


113 


with respect to disposable parameters in E{X) 
and a prescribed weighting function Wo(t,X). 
The lower limit of the integral is indicated as 
— 00 in compliance with the physical impossi- 
bility of discriminating between relevant and 
irrelevant data, with fixed linear networks, ex- 
cept on the basis of age. The burden of dis- 
crimination must be relegated to the weighting 
function which must be a function only of the 
age t ~ k. Under the ideal restriction that 
Wo(t — X) is identically zero when ^ — A > T or 
X < t — Ty the indicated lower limit of the in- 
tegral is purely nominal. 

As in Section 10.2, it is convenient to con- 
duct the analysis in terms of the age variable 
T = t — X introduced there. If 

F(t) = E(X) F(r) = E(X) 

the integral to be minimized may be expressed 
in the form 

[F{t) - F{r)? WM dr . (1) 

In accordance with the discussion of quasi- 
distortionless transmission networks in Section 
A.IO, Appendix A, the smooth curve E (x) 
should be a polynomial in A. Hence F{r) 
should be a polynomial in r. It will be more 
convenient, however, to express F(t) formally 
as a linear combination of polynomials in r 
which may be orthogonalized. Hence, let 

F(r) = Fo+ Fi • G,(t) + y, • U.(r)+ - + y. • GM 

(2) 

where Gmir) is an mth degree polynomial in r. 

Let Wq{t) be normalized in the sense that 

Woir) dr = 1 




In terms of the forward time A, (2) and (3) 
reduce to 

S(A) = yoW + FiW • G,{t - X) + v,{t) • G^it - X) 
+ ”• + y n{t) ‘ Gn{t—X) (4) 

where 

Vm{t) = km I E(x) . Gm(t — X). Wo{t — X)dX. (5) 

Expression (5) identifies the Vm{t) as the 
responses to ^'(a) of fixed linear networks 
whose impulsive admittances are 

Wm(r) = kmGm(r) : WM . ( 6 ) 

By (4), the predicted value may be obtained 
by a linear combination of the responses of 
these networks, viz., 

E{t + tf) = Voit) + G,(~tf) • V,(t) -h G,(-tf) • V,it) 
+ '--\-Gn(-tf) 'Vn{t) . (7) 

A schematic representation of an nth order 
smoothing and prediction circuit, based on (7), 
is shown in Figure 1, where the Gm( —tf) are 
represented as potentiometer factors dependent 
on the time of flight. 


E(t)- 


-I Yi(P)[-AAA^V^^^ 


Y„(p) 


’ 

- \ Y^(p) 


A/VW- >E(t+t^) 


n t - 


Figure 1. Schematic representation of nth order 
smoothing and prediction circuit. 


and the Gmir) be orthogonalized with respect 
to the weighting function W^G) in the sense 
that 


^CD 

J 


Gi{r) GM) W,{t) dr = 0 
1 


(Go = 1, fco = 1). 


if Z m 
if Z = m 


The integral (1) is then a minimum with 
respect to the y„»’s in (2) if 

F(r) ■ GUr) ■ W,{t) dr . (3) 


Alternatively, (7) may be written 

E{t -f tf) = E{t) + [Gi(-Z/)-Gi(0)] • FiCO + ••• 
+ [Gn(-tf)-Gnm 'Vn{t) (8) 

where E (t) is then replaced by E (t) when 
position data smoothing is to be omitted. 

It is not necessary that the GmG) polyno- 
mials be orthogonal. However, the circuit 
switching required to reduce or increase the 
order of the prediction is simplest when the 
GmG) polynomials are orthogonal. Orthogonal 
polynomials corresponding to any prescribed 


confidentiaH 


114 


SMOOTHING FUNCTIONS FOR POLYNOMIAL EXPANSIONS 


weighting function Wq{t) are readily derived 
by well-known methods. 

The weighting function W^ir) may be deter- 
mined by either of the methods described in 
Appendix B as the best weighting function for 
smoothing position data, under prescribed 
tracking error characteristics. Then the best 
impulsive admittances W^nir) for a smoothing 
and prediction circuit, are prescribed by (6). 

The relationship (6) shows that if the pre- 
scribed weighting function Wq{t) satisfies the 
formal requirements for physical realizability, 
so will all of the impulsive admittances Wmir). 
Of the standard sets of orthogonal polynomials 
those of Laguerre appear to be the best adapted 
to physical realization. The Laguerre polyno- 
mials (t) are orthogonal in 0 < t < oo 

with the weighting function However, 

such a weighting function is, in general, very 
unsatisfactory from the practical point of view 
of settling characteristics. 

It is possible of course to approximate any 
prescribed weighting function W^ir) as closely 
as may be desired in a physically realizable 
form, derive a set of orthogonal polynomials 
based on the approximate form, and determine 
the impulsive admittances W^nir) from (6). 
However, such a procedure leads to complexities 
of network configuration which increase very 
rapidly with the index m. This increasing com- 
plexity is hardly justifiable in practice. 

From the foregoing considerations, it ap- 
pears that the most practical procedure is to 
derive all of the impulsive admittances W^(t) 
without regard to physical realizability, ap- 
proximate them independently in physically 
realizable forms of independently prescribed 
complexities, and modify or redetermine the 
potentiometer factors in accordance with the 
discussion in Section A. 10, Appendix A. 


WEIGHTING FUNCTIONS FOR 
DERIVATIVES 


because, with the exception of the 

Wmir ) , as will presently be seen, cannot be nor- 
malized. The term weighting function is re- 
served for the functions defined by (11) below. 

Since t’' is a linear combination of the Ga(r) 
where s = 0, 1, • • • , r, it is obvious from (6) 
that 


T^WUr) dr = 0 

when r < m . 

In particular 



W^nir) rfr = 0 

when m > 0 . 

Since the transmission function Ym(p) of a 
network is the Laplace transform of its im- 
pulsive admittance (see Section A.3), we have 

Ym{p) = / Wmir) 6-P^ dr 

^ {^p)r 

= f- / r'- (r) dr . (9) 

r=o r\ Jo 

The first m terms in this series vanish. Hence 
Ym ip) will be of the form 

Ym{p) = v^ymip) (10) 

where Vmi^) ^ 0. This permits us to regard the 
network whose impulsive admittance is W^ir) 
as an instantaneous mth order differentiator, 
corresponding to the factor p^*^ in (10), in 
tandem with a purely smoothing network 
whose transmission function is Vmip)- 

It is convenient to associate a weighting 
function Wmir) with the purely smoothing net- 
work whose transmission function is Vmip) • 
Dividing (10) through by p^ the resulting 
operationaL equation may be interpreted (see 
Section A.5) to mean that the weighting func- 
tion Wmir) is the m-fold integral of the im- 
pulsive admittance W^ir) between the limits 
0 and r. This is expressed by 



The impulsive admittances defined by (6) 
for m > 0 may not be regarded as weighting 
functions even though the response of the cor- 
responding networks to EM is, by (5) 


Wmir) = ( I Wmir) • . (11) 

Jo Jo 

By a relationship similar to (9) between 2/m (p) 
and WriJr ) , it follows from (0) ^0 that 


F,n (0 = f E(t - r) • Wm (r) - dr, f 

Jo Jo 

(confideatial" 


Wm(r) dr 0 . 


LEGENDRE POLYNOMIALS 


115 


Hence the Wmir) may be normalized in the 
sense that 


Wm (r) dr = 1 

for all values of m. However, this may be done 
in general only if the G„i(t) polynomials are 
not normalized in the sense that = 1 for any 
value of m > 0. It is in fact readily shown that 
the coefficient of r’” in G„i(t) must be the same 
as that of r'” in e-’’. 



it is readily determined that 


1 



[Gm{r)]‘^ Wo(t) dr 


_ ( m !)2 

( 2 m)! ( 2 m + 1 )! ’ 

Then, by (6) 

=(-)m Pm ( 2 r - 1 ) 

= 0 r > 1 . 


0 < r < 1 


LEGENDRE POLYNOMIALS 


Gmir) = i-r 


ml 

(2m) I 


Pm( 2 T — 1) . 


The first few of these are tabulated below. 


m 

0 

1 

2 

3 


Gm(r) 

1 

1 

2 

J III! 

12 2 2 


L. 4. ^ _ I, 

120 10 ^ 4 6 


With the help of the formula 


/; 


[Pm (X)]2 dx 


2 m + 1 


Substituting this in turn into ( 11 ) and making 
use of Rodrigues’ formula 


The Legendre polynomials P,„( a;) are orthog- 
onal with respect to the range — 1 < a; < 1 and 
uniform weighting. In other words, the poly- or 
nomials Pm{ 2 r — 1) are orthogonal with respect 
to the range 0 < r < 00 and the weighting func- 
tion® 


° The unit of time being equal to the nominal smooth- 
ing time. 




( — '\m d^ 


Wo(t) = 1 when 0 < r < 1 

= 0 when r > 1 . 

It is known from Section 10.4 that this form 
for the weighting function Wq(t) is best in 
case the tracking errors are flat random noise. 
In the integral ( 1 ) to be minimized, the G,„(t) 
polynomials should then be 


it is finally found that 
( 2 m -h 1)1 


Wm(T) = 


(m !)2 
0 T > 1. 


[t( 1 - r)]'^ 


0 < T < 1 
( 12 ) 


By a relationship of the form of ( 9 ) the 
transmission functions Vm iv) corresponding to 
the weighting functions may be deter- 

mined. The first three are 


Voiv) = 


1 - e-P 


2/i(p) = -3 [(P - 2) + (p + 2)e-P] 
y^(p) = p [(p2 - + 12) - (p2 + 6 p + 12)e-P]. 


These may be written in the form 

Vmiv) = Qm(co) • 

where 


(13) 


Oo(t^) 


sin X 
X 


(”i) 


3 sin X — X cos x 


Qi(«) = 

JU 

^ ^ - (3 - a;2) sin a: - 3a: cos x . 

.Q 2 (co) = 15 -5 (14) 


; CONFIDENTIAL 




116 

or in the 

yo(p) 

yiip) 


SMOOTHING FUNCTIONS FOR POLYNOMIAL EXPANSIONS 


infinite power-series form 


00 


I 


(-p)” 

(n + 1)! 



n=0 


n + 1 
(n + 3)! 


(-p)” 


p.(p) = 60 f (- p)V (15) 

Methods for obtaining physically realizable ap- 
proximations to the weighting functions Wmir) 
or impulsive admittances based upon 

the Q functions (14) and the series expansions 
(15) are described in Chapter 12. 


Chapter 12 


PHYSICAL REALIZATION OF DATA-SMOOTHING FUNCTIONS 


T his chapter will be devoted to a brief re- 
view of some of the methods and techniques 
which have been used in the physical realiza- 
tion of data-smoothing or weighting functions. 
The first two sections will be devoted to meth- 
ods for determining physically realizable ap- 
proximations to a desired weighting function. 
The third section takes up the use of feedback 
amplifiers and servomechanisms in order to 
avoid the use of coils of generally fantastic 
sizes. The final section takes up the design of 
resistance-capacitance networks. 

Methods of deriving physically realizable ap- 
proximations of best weighting functions may 
be divided into two classes, which may be 
called, for convenience, ^-methods and p-meth- 
ods. The ^-methods are those in which a pre- 
scribed best weighting function W (t) is 
approximated directly by a function Wa(t) of 
realizable form, viz., a sum of decaying expo- 
nential terms and exponentially decaying sinu- 
soidal terms. However, the ^-methods are most 
useful when the approximation is restricted to 
a sum only of exponential terms. According to 
the discussion in Section A.9, Appendix A, such 
a restriction corresponds physically to passive 
RC transmission networks. A ^-method was 
used by Phillips and Weiss in the reference 
quoted in Section 10.3 to obtain an approxi- 
mation with one decaying exponential term and 
one exponentially decaying sinusoidal term. 
However, this method rapidly becomes un- 
wieldy as the number of terms is increased. 

The p-methods are those in which the ap- 
proximation is derived indirectly from the 
transmission function Y (p) corresponding to 
W{t). A rational function Ya(p) approximat- 
ing y (p) is first determined. If it is realizable, 
and it usually is, then Wait) = L-^lYaip)]. In 
general, Yaip) will have complex poles and, 
therefore. Wait) will have exponentially decay- 
ing sinusoids as well as simple exponentials. 
This gives the p-methods a considerable advan- 
tage over the ^-methods in more efficient use of 
network elements. The fact that this generally 
calls for impractical element values in passive 


RLC networks is not serious. As shown in Sec- 
tion 12.3, the use of coils may be avoided 
entirely by the use of feedback amplifiers. 

^-METHODS 

To describe the f-method,^ let 

Wa(t) = + . • ■ + Ae^-’rf (1) 

where the a’s are prescribed and the A’s are to 
be determined. Two considerations are involved 
in the determination of the A’s. The first con- 
sideration is based on the relationship between 
the continuity conditions at ^ = 0 and the ulti- 
mate slope of the loss characteristic as ex- 
pressed in the theorem in Section A.8. Accord- 
ingly, a number of relations of the type 

Ai + A 2 + • • • -f- An = 0 

a\ Ai -|- 0:2 A 2 + • • • + 0!n An = 0 (2) 

Ai -h o ;2 A 2 + • • • + An = 0 r < n — 1 

must be satisfied. This leaves n — r — 1 oi the 
A's for the second consideration. 

The second consideration concerns the man- 
ner in which the approximation in the range 
i > 0 is to be made. The approximation may, 
for example, be required to pass through 
n — r — 1 points on W (0 or, the first n — r — 1 
moments of the approximation may be required 
to be equal to the corresponding moments of 
Wit), The latter is expressed by relations of 
the type 

Ai A 2 An 1 r 

— + T + • " + “7 = ^ / ^(0 dt 

s = 1, 2, • . . , n — r — 1 (3) 

Foster’s investigations were concerned only 
with the parabolic weighting function (4) 
Chapter 10, so that only the first of (2) was 
involved. Numerical studies led to the belief 
that, with a given number of a’s, the best ap- 
proximation was to be had from the case in 


“ The i-method is principally due to R. M. Foster. 




CONFiCl 


117 


118 


PHYSICAL REALIZATION OF DATA-SMOOTHING FUNCTIONS 


which all of the a’s are equal. Hence the natural 
center of attention was the special form 

Wait) = (A.t + + . • . + (4) 

At large values of t this expression reduces ap- 
proximately to the last term, and if it is as- 
sumed that An-i = 1, the settling condition fixes 
a to at least a first approximation. The rest of 
the work of approximating the parabola is then 
equivalent to a problem in polynomial approxi- 
mation. Once the A’s are determined, a better 
value of a can be found from the settling con- 
dition, and the process gone through again. 

If the a’s are only approximately equal, the 
approximation will still behave approximately 
like (4) with an average value used for a. The 
difficulty with equal or nearly equal a’s is that 
it leads to networks with extreme element 
values. In order to secure satisfactory element 
values, it is generally necessary to depart sub- 
stantially from the condition of equal a’s. This 
results in some, but not a large, loss of effi- 
ciency in approximating the parabola. Foster 
recommends that the a’s be chosen as a geo- 
metric series, with their geometric mean more 
or less around the equivalent point for equal 
a’s. With four a's he suggests that the constant 
ratio in the series may be 3:2, whereas with 
only two a’s the ratio should be raised to 2:1. 
These are, however, only rough values and 
obviously depend on individual opinion of what 
constitutes an unreasonable element value. 

As a matter of experience, it turns out that 
the characteristic first obtained usually has a 
rather long and slowly decaying tail, as shown 
in Figure 1. This, of course, is equivalent to a 



Figure 1. Approximation to parabolic weighting 
function, showing poor settling characteristic. 

correspondingly long ‘‘settling time,” or time 
before a useful prediction can be made. In 
practice, therefore, after the preliminary 
design has been found, adjustments are made 
to bring the tail of the curve under control. 


partly by modifying the values of the A’s 
slightly, and partly by contracting the time 
scale to bring the part of the tail which remains 
appreciable within the allowable settling time 
limits. This leads to the somewhat lopsided 
match to the parabola shown in Figure 2. 



Figure 2. Approximation to parabolic weighting 
function, showing better settling characteristic. 

A method of bringing the tail of the curve 
under controP is to minimize the expression 

[Wamdt = 22 ClmAiAm (5) 

l,m=l 

where 


under the restrictions (2) and all but the last 
of (3). 

The t-method used by Phillips and Weiss is 
based on a 3-term approximation of the form 
(1) in which one a is real while the other two 
may be conjugate complex. The a’s are not 
prescribed, so that there are six parameters to 
be determined. Four restrictions are imposed, 
viz., the first of (2), the first of (3), a restric- 
tion on the value of the tail area, viz., 

Wait) dt = h , 

^=1 ai 

and the cross-over condition 

WaiT) = 0 . 

Finally, the transmitted noise power, which, 
under the assumption of fiat random noise as- 
sociated with the position data, takes the form 
(see Section 10.4) 

” IWait)]^ dt 

0 

is minimized with respect to the two remaining 
parameters by numerical methods. 

»> Used by R. F. Wick. 





/j-METHODS 


119 


122 20-METHODS 

Three p-methods have been used. These will 
be described in chronological order. 

The first p-method is one which was used by 
R. L. Dietzold in exploiting the use of feedback 
amplifiers to secure the advantages of approxi- 
mations with complex exponentials. The trans- 
mission function F(p) corresponding to the 
best weighting function W {t) is first formu- 
lated. The loss characteristic, —20 logjo \ Y (iw) |, 
is next computed and plotted against the fre- 
quency on a logarithmic scale. Then standard 
equalizer design techniques are employed to ap- 
proximate the loss characteristic, keeping in 
mind that the transmission loss in the feedback 
network of a feedback amplifier becomes a 
transmission gain for the circuit as a whole 
(see Section 12.3). 

The second p-method is merely a more com- 
plete analytic formulation of the first, thereby 
avoiding the necessity for employing equalizer 
design techniques. It depends upon the possi- 
bility of expressing the transmission function 
corresponding to the best weighting function, 
in the form of equation (13) Chapter 11, which 
is associated with the symmetry of the weight- 
ing function, as shown in Section A.7. The 
method is based upon the determination of the 
envelope of the Q-function. The Q-function is 
first differentiated in order to obtain the 
equation which determines the values of w 
at which the maxima and minima occur. This 
transcendental equation is not solved but is 
used to eliminate the trigonometric functions 
in the expression of the Q-function. The result- 
ing expression, which is an irrational function 
of ( 0 ^, is then squared in order to make it a 
rational function of The substitution 

= — 0 )^ is made and the expression is then re- 
solved into two factors of which one contains 
all the poles with negative real parts while the 
other contains all the poles with positive real 
parts, the two factors being conjugate complex 
when p = iw. The first factor is then taken as an 
approximation of the desired transmission 
function. Applying the method to the desired 
transmission functions defined by (13) and 


(14) of Chapter 11, we get 


/ ^ ^ 12 

12 + 6p + 

^ 120 + 60p + 12p2 + ® 

This last is the basis for the design of a posi- 
tion and rate smoothing circuit for a proposed 
computer for controlling bombers from the 
ground.^^’^2 This design is described briefly 
in Chapter 13. 

The third p-method is based upon the ascend- 
ing power-series expansion of the transmission 
function corresponding to the best weighting 
function. Examples of such power series are 
given by (15) of Chapter 11. The method of 
approximation is one which is credited to Fade 
in 0. Perron’s “Kettenbriichen.”^® If the discus- 
sion in Section A.8 is referred to, it will be seen 
to be also a method of moments. 

The method consists in determining the co- 
efficients in a rational function of the form 

1 + QlP + Ct2p2 + • • • + amP''' /yN 

1 + felP + ^2p2 + • • • + tnP” 

SO that the ascending power-series expansion 
of the rational function will agree with that of 
the best transmission function, term for term 
up to and including p”*+”. If the series for the 
best transmission function is 

1 + CiP -I- C2p2 + • • • + Cm+nP^'^^ + • • • (8) 

the equations which determine the coefficients in 
(7) are obtained by equating coefficients of 
corresponding powers of p, up to and including 
the (m + n)th, in 

(1 + 6lP + • • • + &nP”) (1 + CiP -h • • • 

and 

1 + cqp + • • • + 

The last n equations will be homogeneous in 
the h’s and c’s. 

It has been expedient in some cases to omit 
the last few of the (m+n) equations in order 
to have some control over the number of real 
roots and poles and the number of conjugate 
pairs of complex roots and poles in the result- 
ing rational function. 

In the assumed rational expression (7) the 



120 


PHYSICAL REALIZATION OF DATA-SMOOTHING FUNCTIONS 


difference n — m should be chosen so that the 
ultimate slope of the loss characteristic will be 
the same as for the best transmission function. 
According to the theorem in Section A. 8, if 
W{t) behaves like as ^ 0, we should take 
r?. — m==r + l. Asa matter of experience the 
rational expression has invariably turned out 
to be physically realizable whenever this “rule” 
was followed. Frequently, however, the rational 
expression has turned out to be physically 
realizable under small departures from the 
rule. 

Examples of this method are given in Chap- 
ter 13. 


age V {t) across the input terminals of the 
second network were in fact under the control 
of the current through the conductor, as shown 
schematically in Figure 5, in such a manner 



Y| 


rl 


Yz 


•(|)V 









Figure 5. Output voitage controlled by short- 
circuit current across intermediate terminals. 


that it had to develop that voltage V {t) which 
reduces the current in the conductor to zero, 
then 


Fi(p) . E(t) + Y,{v) . V{t) = 0 . 


12 3 USE OF FEEDBACK AMPLIFIERS 
AND SERVOMECHANISMS 

In this section we shall describe the use of 
feedback amplifiers and servomechanisms to 
obtain desired transmission functions. For com- 
plete discussions of the most recent technical 
advances in the analysis and design of feedback 
amplifiers and servomechanisms the reader 
should consult some of the modern literature 
on these subjects.^*®'®'!®’!®'!^ 

Let us assume that we have two networks 
whose transmission functions are Y^{p) and 
F 2 (p), respectively, as shown in Figure 3. For 


E(t) 



I,(t) 



I,(t)=Y,(p)-E(t) l2Ct) = Y2(p)-V(t) 

Figure 3. Schematic representation of networks 
intended for feedback circuit application. 


a signal E{t) applied to the first network the 
short-circuit output current is E{t) = Fi(p)* 
E{t). For a signal V {t) applied to the second 
network the short-circuit output current is 



Figure 4. First step in combining networks. 


Izit) = Fi (p) *7(0 . With the networks sharing 
a common short-circuiting conductor as shown 
in Figure 4, the current through the conductor 
is /i + /g. If the source which develops the volt- 


Hence, the transmission function (now a volt- 
age-voltage ratio) of the arrangement shown 
in Figure 5 must be 

This relationship provides a method of ob- 
taining transmission functions with complex 
poles without the requirement of coils.® The 
complex roots of F (p) , must be assigned to the 
numerator of Y^{p), and the complex poles of 
F(p) to the numerator of Y^iv)- Aside from 
this, the other roots and poles of F (p) may be 
assigned in any way which is favorable to good 
design practice. Redundant factors may be in- 
troduced if they are desirable, as is done in the 
examples described in Sections 13.1.5 and 13.3. 

The source of the voltage V {t) in Figure 5 
does not have to be controlled by the current 
through the short-circuiting conductor. Since 
the current through any short circuit must be 
zero if the voltage across the short-circuited 
terminals is zero before the short circuit is con- 
nected across them, the source of the voltage 
V {t) may just as well be controlled by the 
open-circuit voltage, as shown in Figure 6. It 
is clear that the source of the voltage V (t) is 
ideally an infinite gain amplifier. It is not nec- 
essary, however, that the amplifier have ideally 
unilateral transmission and infinite input and 
output impedances, since departures from these 
ideal characteristics may be compensated for in 
the design of the feedback network. 

The simple result expressed by (9) may be 
readily modified to take account of the finite 


® This observation was first made by R. L. Dietzold. 


CONFIDEXTIAL 


J 





DESIGN OF RC NETWORKS 


121 


gain of a physical amplifier. The modification 
will be expressed as an extra factor which 
corresponds to the effect"’ or error”^®® 
commonly encountered in the theory and design 
of feedback amplifiers. 



Figure 6. Output voltage controlled by open- 
circuit voltage across intermediate terminals. 


The exact transmission function of the cir- 
cuit shown in Figure 6 is most simply ex- 
pressed in terms of the following quantities: 
Fi(p) = current through a short across ter- 
minal-pair No. 3, per unit emf applied 
across terminal-pair No. 1. 

FaCp) = current through a short across ter- 
minal-pair No. 3, per unit emf applied 
across terminal-pair No. 2. 

-^2 (P) == impedance between terminal-pair No. 

2, with terminal-pair No. 3 shorted. 

Z3 (p) = impedance between terminal-pair No. 

3, with amplifier dead, terminal-pair 
No. 1 shorted, and terminal-pair No. 2 
open. 

G(p) = transadmittance of amplifier. 

Then 


Y = 




GF2Z2Z3 


(10) 


The quantity GYoZ.Z^ is the /x/? of the circuit. 
The quantity Y^Y^Z.,Z^ to which Y reduces 
when G = 0 represents the direct transmission 
of the circuit. 

The active impedance across terminal-pair 
No. 2 is 


where 


Z2A 


Z2P 

GF2Z2Z3 


( 11 ) 


Z 2 P = Z2(l + YIZ,Z,) . .(12) 

Z 2 P is the passive impedance across terminal- 
pair No. 2. It differs from Z^ in that terminal- 
pair No. 3 is open. 


The exact expression (10) of the transmis- 
sion function is useful chiefly as a check on the 
simpler but approximate expression (9). It is 
in general quite practicable to make the trans- 
admittance or transconductance G of the am- 
plifier large enough so that the effect may be 
neglected. 

In accordance with the sense in which the 
term “servomechanism” is used by MacColl,^ 
a feedback circuit, such as that shown in Fig- 
ure 6, is a servomechanism — more specifically, 
an electronic servomechanism — since it oper- 
ates on the ideal principle of maintaining zero 
voltage across the terminal-pair No. 3. An 
electromechanical counterpart of the circuit 
shown in Figure 6 is shown in Figure 7. These 


2- PHASE INDUCTION 
MODULATOR MOTOR 



Figure 7. Electromechanical counterpart of feed- 
back amplifier circuit resulting in servomechanism. 


circuits assume that the signal E{t) is a modu- 
lated d-c carrier. 

If the signal is a modulated a-c carrier, 
“shaping” cannot be done conveniently by elec- 
trical networks. The difficulty may be avoided 
by various special devices. An example is de- 
scribed and illustrated in Section 13.4. 

12^ DESIGN OF RC NETWORKS 

In this section we will describe and illustrate 
two general methods of designing RC networks. 
The first is most useful when the transmission 
function is finite and not zero at zero fre- 
quency ; the second, when the transmission 


^CONFIDENTIAL^ 


122 


PHYSICAL REALIZATION OF DATA-SMOOTHING FUNCTIONS 


function is zero at zero frequency. The case of a 
transmission function with a pole at zero fre- 
quency will not be considered, since it is cov- 
ered by the methods described in the preceding 
section, in conjunction with the methods de- 
scribed below. 

Let 


Y{v) 


up ~f~ (L\P ^ 

1 + felP + • • • + hnV"' 


{a, > 0) (13) 


with simple, real, negative poles. Dividing by 
p, expanding into partial fractions and multi- 
plying through by p, we get 


Y{v) = + (ao + 

Un 


pAi pA^ _ 

( pBi , pB2 . 

\P + ^1~^ P + ^2^ ' " 


where the A’s, B's, a’s and ^’s are positive real 
quantities. The first term must be associated 
with those in the first parentheses if a^+i > 0, 
with those in the second parentheses if an+i < 0. 
The transmission function is now in the form 


Y{p)=Ya{p)-Yb{p) (14) 

where Ya(p) and Yeip) are physically real- 
izable driving-point admittances of RC type. 
Each term of the form pA/{p + a) is the admit- 
tance of the two-terminal, two-element network 


The transmission function (14) may be real- 
ized in the arrangement shown in Figure 9 
or in that shown in Figure 10. The latter is 
a lattice network which is suitable only in a 



Figure 10. Lattice prototype for passive net- 
works with RC transmission characteristics. 


balanced-to-ground circuit. To obtain an un- 
balanced passive equivalent of this network we 
may resort to steps which will be described 
later in this section. 

The second general method of designing RC 
networks is most useful when 


Y(7i) = ^ ^0 + + • • • + (^nP^ 

^ 1 + 6iP + • • . + bnP^ 


(Go > 0) (15) 


with simple, real, negative poles. Now, if the 
lattice in Figure 10 were driven from an in- 
finite-impedance source of current the out- 
put current would be 


R = “ C“ — 


D — ^AA^\ — 11 — o 

Figure 8. Simple RC network. 


shown in Figure 8. Each term in (14) there- 
fore represents a parallel combination of two- 
element networks of the type shown in Figure 
8 and a conductance o-o in the case of Ya(p), 


PHA8F ) 


= (Ya-Y»)-E 


PHASER 

INVERTER 


SUMMING 

AMPLIFIER 


Figure 9. Method of realizing RC transmission 
functions, requiring phase inverter. 


and a capacitance \an+i\/bn in the case of either 
Y a{p) or Yb{p). By well-known methods these 
two-terminal networks may be transformed 
into a variety of other configurations. 


/ 


If, furthermore. 


Ya 


then 



Taking it for granted for the moment that the 
lattice can be transformed as shown schemat- 
ically in Figure 11, we may then discard the 
condenser across the output terminals and, by 
Thevenin’s theorem, we may replace the 
condenser across the input terminals and the 
infinite-impedance current source by a series 
condenser and a zero-impedance voltage source. 
The result is shown in Figure 12. Since 


1 - — 
Ya 


h 


p 

p 


(16) 


CTNFIDENTTA^ 


DESIGN OF RC NETWORKS 


123 


h — pCoE we now have 

T _ ^ V F 
k ' 

which is the desired result, to a constant factor. 

The factor k should in general be taken as 
small as possible subject to the requirement 
that all the roots and poles of (16) be simple, 





Figure 11. Step in transformation of networks 
with zero transmission at zero frequency. 


3. Series network pulled out of both branches : 
shown in Figure 15.® 

4. Series network pulled out of the lattice 
branch only : shown in Figure 16.® 



Figure 13. Step in transiormation oi lattice; 
shunt networks pulled out of both branches. 


real, and negative. It can always be taken large 
enough to fulfill this requirement. A suitable 
value may be easily chosen by inspection of a 
plot of Y (p) /p for negative real values of p. 



Figure 12. Final step in transformation of net- 
works with zero transmission at zero frequency. 

The numerator and denominator of (16) are 
of equal degree and therefore contain the same 
number of linear factors. These factors may be 
assigned to Ya or to Yb arbitrarily except that 
Ya and Yb must be physically realizable driv- 
ing-point admittance functions which behave 
ultimately like condensers as the frequency in- 
creases indefinitely; that is, roots and poles 
must alternate and there must be a simple pole 
at infinity. 

There are five kinds of steps which may be 
taken to transform a lattice into an unbalanced 
form. These steps are based upon Bartlett's 
bisection theorem,^^ and may be taken in any 
order and as often as necessary. Each of them 
will now be described as it would be applied 
directly to Figure 10. In the following diagrams 
a lattice enclosed in a rectangle means an un- 
balanced network whose configuration may not 
be known yet, but whose lattice prototype is as 
indicated. 

1. Shunt network pulled out of both branches : 
shown in Figure 13. 

2. Shunt network pulled out of the line branch 
only: shown in Figure 14. 



Figure 14. Step in transformation of lattice; 
shunt network pulled out of line branch only. 



Figure 15. Step in transformation of lattice; 
series networks pulled out of both branches. 



Figure 16. Step in transformation of lattice; 
series network pulled out of lattice branch only. 

® Given in impedance form. 


124 


PHYSICAL REALIZATION OF DATA-SMOOTHING FUNCTIONS 


5. Breakdown into parallel lattices: a fairly 
obvious step which need not be illustrated. 

As an example of (13) consider 


Y(p) 


dp + aiP + 02V^ 

1 + bip 

where all the coefficients are positive. Since 


Y{p) + flo - 


(apbi — aibi -f- 02)p 

«(>■ + 1 ) 


there is no problem if > (a^/b^) + But if 
di < (ao/bj + ttohi we have the problem of trans- 



Figure 17. 


Ci--^ G|=^o 

_ apbf- a|b| -i-ag 

4“ b, 




apbi — a|b| + 


Illustrative lattice prototype. 


forming the lattice in Figure 17. We can apply 
steps 2 and 4 immediately, but find that the 
residual lattice cannot be transformed unless 
0^1 > («' 2 /^i)- Under this additional restriction 
we can apply step 3 obtaining finally the net- 
work shown in Figure 18. 

As an example of (15) consider 

Y(p) = p ^ . 

Taking A: = 1 (the smallest value which may be 
assigned), we get 

Yb ^ 2p(3 + 16p) 

Ya (1 + 2p) (1 + 16p) • 

One way of choosing Ya and is 


^ (1 + 2p) (1 -h 16p) 

^ 2(3 + 16p) 


Yb = p. 


This leads finally to the network shown in Fig- 
ure 19. Such a simple network is possible of 


course because Y (p) happens to satisfy the re- 
quirements of a physically realizable driving- 
point admittance function. However, another 
way of choosing Ya and is 

_ 1 + 2p _ p(S + 16p) 

2 1 + 16p 

This leads to the network shown in Figure 20. 



Figure 18. Unbalanced equivalent of illustrative 
lattice prototype when a. 2 /Si<a-i< (a 2 /^i) + apbi. 



Figure ly. kC network with zero transmission at 
zero frequency. 



Figure ZU. Another RC network witn zero trans- 
mission at zero frequency. 


Chapter 13 

ILLUSTRATIVE DESIGNS AND PERFORMANCE ANALYSIS 


T he illustrative material described in this 
chapter is taken from four practical appli- 
cations. 

1. Second-derivative circuit for the M9 anti- 
aircraft director. 

2. Position data smoother for the ‘‘close sup- 
port plotting board,” with delay correction for 
constant velocity aircraft. 

3. Position and rate circuit for the “com- 
puter for controlling bombers from the 
ground,” with optional delay correction of posi- 
tion data for constant-velocity aircraft. 

4. Position and rate circuit using electro- 
mechanical servomechanisms. 

The design and analytical procedure used in 
the first application has not heretofore been 
described in writing. Hence, considerably more 
space will be devoted to it than to the other 
three applications. The latter have been de- 
scribed in detail in reports.^^-^^.is 


should cut off at the rate of 12 db per octave.® 
This requirement was set as a precaution 
against noise due to granularity of the coordi- 
nate-conversion potentiometers in the director. 

Following the procedure outlined in Section 
12.2 the following equations were obtained : 


6, - I = 0 

?>2 — 2 + y 

63 + = 0 

64 - 2 63 + y 62 ~ * * *** + 1008 ^ ° 

whence 



Since 


62 


3 7. - i ^ _ 1 

28’ 84’ 1764 ‘ 


+ 21p3 + I89p2 + 882p + 1764 

21 + 

2 


= 


P + 42) 


13 1 SECOND-DERIVATIVE CIRCUIT 
DESIGN 

Realizable Approximation of Best 
Transmission Function 

The best transmission function for the sec- 
ond-derivative circuit was taken to be 

F2(p) = P2^2(P) , 

in the notation of Chapter 11. This assumes flat 
random noise in position data and, arbitrarily, 
1-second smoothing and settling time. The 
series expansion of y^iv) is, according to ex- 
pressions (15) of Chapter 11, 

y^{v) = i - \v + \v^~ + 

The form of the rational approximation, 

yiv) - I _|_ ^2^2 -f- 53p3 

was chosen for simplicity under the require- 
ment that the transmission function p^y(p) 


X (p2 + + 42) , 

2/2 (P) would have two conjugate pairs of com- 
plex poles, viz., 

p = - 6.40 ±: il.047, - 4.10 ± i5.02, 

of which one pair is very nearly real. 

In order to simplify the circuit design, how- 
ever, it was desirable to limit the number of 
complex poles to a single conjugate pair. This 
was accomplished by leaving 64 arbitrary so 
that the denominator of yziP) was 

^ + + + + • 

A value for which would make this expres- 
sion vanish at two negative real values of p 
was found by plotting 

176464 = ^ (x^ - 9x2 -1- 42x - 84) 

* The design antedated the formulation of the n — m 

= r + 1 rule given in Section 12.2, according to which 
the best transmission function should have been taken 
as p^ys(p) in the notation of Chapter 11. However, no 
trouble was experienced in obtaining a physically real- 
izable approximation, of the complexity assumed. 

125 


/CONFIDENTIAL 


126 


ILLUSTRATIVE DESIGNS AND PERFORMANCE ANALYSIS 


against x, as shown in Figure 1. The right- 
hand member is positive only in the range 
X > 3.77 and has a maximum of 0.982 at about 
X = 6.63. 


It has been noted that Figure 3 also repre- 
sents the best and the realized weighting func- 
tions. 



1.0 2.0 4.0 6.0 8.0 10.0 

Figure 1. Graphical determination of 64 . 


In order to obtain a substantial separation 
between the two real poles of the value 

176464 = 0.5 was chosen. The approximation 


i/(p) 


^ ■*" 2 ^ 28 84 3528 


has poles at 

p = - 4.17391 , - 31.72813 , - 3.04898 

± i 4.16463 . 

The series expansion of 1/2 (p) agrees with that 
of 2/2 (P) to four terms, the fifth term being 
37/7056 instead of 5/1008 p^ The difference 
in the fifth term is less than 6 per cent. 

The realized approximation and the best 
weighting function are shown in Figure 3. 



Figure 2. Responses to step function, viz., E{t) = 
1 when t > 0 . 



% 

Figure 3. Responses to linear ramp function, viz., 
E(t) = t when t > 0; second derivative smoothing 
functions. 


Transient Responses 

The responses of the physical network whose 
transmission function is P^ 2 / 2 (P) compared 
to those of the best network whose transmis- 
sion function is p^Pa (P) > in Figures 2, 3, and 4. 
The signals for which (and the formulas by 
which) these responses were computed are 
tabulated below. 

Figure Signal 

« < 0 f > 0 
2 0 1 

3 0 t 

4 0 -P 

2 


Response formulas 
Realized Best 

L~'^[py2(p)] 60<(1 — 20(1 — <) 

L-^[y2(p)] 30101 - 01 =^ 

A“'[i^ 2 (p)] <’(10-15< + 6<2) 


1.2 

1.0 

0.S 

0.6 

0.4 

a2 

0 

Figure 4. Responses to parabolic ramp function, 
viz., E(t) = (^)P when t > 0; second derivative 
settling characteristics. 






SECOND-DERIVATIVE CIRCUIT DESIGN 


127 


If a signal of the form 

E { t ) = Qq + a it ^ ^2^2 

were to be applied suddenly to the second-de- 
rivative circuit at ^ = 0 the response would be 

y{t) = ^ , 

where A^, Aj, Ao stand for the responses shown 
in Figures 2, 3, and 4, respectively, and where t 
is the time in seconds and T is the nominal 
smoothing time. The response V (t) is the indi- 
cated acceleration of the target. 

The sudden application of the instantaneous 
position and velocity components of the signal 
to the second-derivative circuit will give rise to 
some very serious consequences unless special 
measures are taken to mitigate them. To see 
this let it be assumed that T = 20 seconds and 
that the target is at such a range that do = 
20,000 yards when the signal E (t) is applied 
to the second-derivative circuit. Each unit of 
Aq in the ordinate scale of Figure 2 then repre- 
sents an indicated acceleration of 50 yd per 
sec^. Referring to Figure 2 it is clear not only 
that the effective settling time will be several 
times the smoothing time but also that the indi- 
cated acceleration will go through exceedingly 
large maxima. 

Exceedingly large transient responses are 
not peculiar to second-derivative circuits. They 
occur also in first-derivative circuits in linear 
prediction, where they are due entirely to the 
initial position term in the signal. In all cases 
they are reduced to harmless proportions by 
special arrangements of the circuits during the 
operation of slewing. 


tion 3^2 of the experimental second-derivative 
circuit design, also referred to a nominal 
smoothing time of 1 second. The transmission 
function of the linear prediction circuit with 
10-second smoothing of first derivative is then 


Table 1 * 


9 / 


Fi 

Y2 




i 


i 

1 

0.174 

0.666 

- 0.454 

0.165 

2 

0.651 

1.166 

— 1.442 

1.212 

3 

1.312 

1.358 

— 2.014 

3.527 

4 

1.943 

1.203 

— 1.069 

6.688 

5 

2.382 

0.821 

2.000 

9.409 

6 

2.599 

0.364 

6.575 

10.115 

7 

2.637 

— 0.067 

10.893 

8.220 

8 

2.558 

— 0.429 

13.468 

4.695 

9 

2.416 

— 0.711 

14.096 

0.953 

10 

2.242 

— 0.920 

13.401 

— 2.092 

11 

2.062 

— 1.070 

12.064 

— 4.320 

12 

1.885 

— 1.172 

10.530 

— 5.777 

13 

1.720 

- 1.238 

9.027 

- 6.704 

14 

1.566 

- 1.279 

7.652 

- 7.169 

15 

1.429 

- 1.299 

6.438 

- 7.398 

16 

1.305 

- 1.304 

5.382 

- 7.446 

17 

1.194 

- 1.299 

4.471 

- 7.374 

18 

1.096 

- 1.286 

3.683 

- 7.221 

19 

1.004 

- 1.268 

3.015 

- 7.025 

20 

0.926 

- 1.247 

2.436 

- 6.795 

22 

0.790 

- 1.198 

1.509 

- 6.292 

24 

0.683 

- 1.145 

0.818 

- 5.780 

26 

0.593 

- 1.091 

0.301 

- 5.287 

28 

0.518 

- 1.040 

0.088 

- 4.826 

30 

0.457 

- 0.991 

- 0.380 

- 4.402 

32 

0.407 

- 0.945 

- 0.599 

- 4.016 

34 

0.364 

- 0.902 

- 0.762 

- 3.666 

36 

0.326 

- 0.862 

- 0.881 

- 3.348 

38 

0.296 

- 0.825 

- 0.967 

- 3.062 

40 

0.266 

- 0.790 

- 1.026 

- 2.800 


* / is in c when smoothing time T = 1 sec. For T-second net- 
works, values of 9/ are multiples of 1/9T c, values of should 
be divided by T, and values of Y 2 should be divided by T^. The 
two networks may have different values of T. 


13.1.3 Effect of Tracking Errors on 
Accuracy of Prediction 
The statistical effect of tracking errors on 
the accuracy of prediction is most readily de- 
termined from the power spectrum of the 
tracking errors and the transmission function 
of the prediction circuit. 

Table 1 gives the values of the transmission 
function of the first-derivative circuit in the 
M9 director, referred to a nominal smoothing 
time of 1 second,*’ and the transmission func- 


while that of the quadratic prediction circuit 
with 20-second smoothing of second derivative 
is 


YM = Yiiv) + 


G 2 • 3^2 (20p) 
400 


where and are determined in accordance 
with the discussion in Section A. 10. Since 


Yiip) = p(l - 0.3724P + . . . ) 
Y2ip) = p2(l ) 

we get 


b 


Yiip) = p 


/ 0.9494 
\p + 1.6 


8.677 34.74 

p + p + 3.6 



Gi = tf 

G 2 = 1 </ + 3.724t/ . 


ILLUSTRATIVE DESIGNS AND PERFORMANCE ANALYSIS 


128 


Table 2 gives the values of \Yi{‘p)\^ and of 
I Yq ip) for tf = 5, 10, 15, 20 seconds. These are 
plotted in Figures 5, 6, 7, and 8. 



Figure 5, Power transmission ratio of linear 
and quadratic prediction circuits with 5-second 
prediction time. 

The last column of Table 2 and Figure 9 
give the power spectrum of a composite of the 
range and transverse errors in a typical run 
made with an experimental Mark VII radar. 
The power contained in the frequency range 
covered by the table accounts for 78 per cent 


of the total power, or an rms error of 15.8 
yards out of 17.9 yards. 

The rms error of prediction is the square 
root of the power transmitted by the prediction 
circuit. This is tabulated on the last line of 
Table 2 and in the smaller table following. 



Figure 6. Power transmission ratio of linear and 
quadratic prediction circuits with 10-second pre- 
diction time. 


Table 2 


90 / 

\yiV 

= 5 


■ o 


15 

\yaV 


20 

\y,\‘‘ 

P*Mk-VII 

0 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

31.4 

1 

1.29 

1.13 

1.82 

1.60 

2.59 

2.71 

3.59 

4.81 

33.5 

2 

2.10 

2.76 

4.08 

8.90 

6.97 

23.16 

10.74 

50.35 

35.7 

3 

3.20 

6.85 

7.19 

26.73 

12.96 

72.51 

20.51 

159.43 

19.7 

4 

4.2 

10.0 

10.1 

39.5 

18.6 

106.1 

29.76 

231.3 

3.6 

5 

5.0 

10.5 

12.1 

39.9 

22.4 

104.4 

35.9 

223.9 

2.5 

6 

5.3 

9.8 

13.1 

35.6 

24.3 

90.6 

38.9 

190.6 

1.2 

7 

5.4 

8.8 

13.2 

30.8 

24.6 

76.6 

39.4 

158.4 

1.6 

8 

5.2 

7.9 

12.8 

26.6 

23.8 

64.7 

38.2 

131.8 

2.1 

9 

5.0 

7.1 

12.2 

23.0 

22.5 

55.0 

36.0 

110.6 

1.4 

10 

4.7 

6.3 

11.4 

20.0 

21.0 

47.0 

33.5 

93.5 

0.7 

11 

4.4 

5.7 

10.5 

17.5 

19.3 

40.4 

30.8 

79.6 

0.8 

12 

4.1 

5.1 

9.7 

15.3 

17.7 

35.0 

28.3 

68.2 

0.8 

13 

3.8 

4.6 

8.9 

13.5 

16.3 

30.4 

25.8 

58.9 

0.5 

14 

3.6 

4.2 

8.2 

12.1 

14.9 

27.1 

23.6 

52.0 

0.3 

15 

3.4 

3.8 

7.6 

10.6 

13.7 

23.4 

21.6 

44.5 

0.8 

16 

3.2 

3.5 

7.0 

9.5 

12.6 

20.6 

19.8 

39.0 

1.1 

17 

3.0 

3.2 

6.5 

8.5 

11.6 

18.3 

18.2 

34.4 

0.8 

18 

2.8 

3.0 

6.0 

7.7 

10.7 

16.3 

16.8 

30.4 

0.4 

19 

2.7 

2.8 

5.6 

7.0 

9.9 

14.6 

15.5 

27.0 

0.7 

20 

2.5 

2.6 

5.3 

6.3 

9.2 

13.1 

14.4 

24.1 

1.0 

rms 

error of 
prediction 

23.9 

29.5 

33.9 

53.4 

44.5 

85.4 

55.4 

125.0 



** P ia in units of 180 yd^ per c. 


CONFIDENTTA 


SECOND-DERIVATIVE CIRCUIT DESIGN 


129 


Time of flight 

Rms error of prediction due 
to tracking errors in yards 

in seconds 

Linear 

Quadratic 

5 

23.9 

29.5 

10 

33.9 

53.4 

15 

44.5 

85.4 

20 

55.4 

125.0 


It is obviously relatively disadvantageous to 
use quadratic prediction when the target is in 
fact flying a rectilinear unaccelerated course. 



Figure 7. Power transmission ratio of linear 
and quadratic prediction circuits with 15-second 
prediction time. 



Figure 8. Power transmission ratio of linear and 
quadratic prediction circuits with 20-second pre- 
diction time. 


The relative advantage of linear prediction 
should persist for target paths with only a 
slight amount of curvature, but this relative 
advantage should decrease as the curvature is 
increased. When the curvature exceeds a cer- 
tain amount, the relative advantage should 
shift to quadratic prediction. 

The determination of the minimum value of 


target path curvature at which quadratic pre- 
diction becomes relatively advantageous de- 
pends not only upon : 

1. dispersion of the predicted point of im- 
pact due to tracking errors, 

but also upon a number of other factors, among 
which are: 

2. actual future position of target with 
respect to the predicted point of impact, assum- 
ing an accurate computer and the absence of all 
sources of dispersion enumerated here 

3. dispersion due to inaccuracies in the com- 
puter and data-transmission systems ; 

4. dispersion due to noise in the computer 
and data-transmission systems ; 

5. dispersion due to variations in actual dead 
time; 

6. dispersion due to gun wear and to varia- 
tions in powder charge, shell weight, shell 
shape, etc.; 












V 

30 















POWER SPECTRUM 

OF 

TRACKING ERRORS 
MARK VII RUNS4SAS4 



20 ^ 
t/> 

o 

15 1 

o 

10 g - 

5 


































0 ^ 










90f *— 

1 |i 

0 1 

2 1 

4 1 

A II 

i io 


Figure 9. Composite power spectrum of tracking 
errors of experimental radar. 


7. dispersion due to variations in meteoro- 
logical conditions along the path of the shell ; 

8. dispersion due to variability of time-fuze 
calibration; and 

9. lethal pattern of shell burst. 

In a special illustrative case, a numerical 
analysis, including most of these factors (esti- 
mated), showed that quadratic prediction be- 
comes relatively advantageous when the target 
acceleration exceeds about O.l^r. However, this 
should not be taken as a general result. 


° This is considered in detail in the next section. 

/CONFIDENT 


130 


ILLUSTRATIVE DESIGNS AND PERFORMANCE ANALYSIS 


13.1.4 Linear and Quadratic Prediction 
Errors on Constant-Velocity 
Circular Courses 

The use of a finite number of derivatives of 
the tracking data for purposes of prediction is 
itself a source of prediction errors even if there 
were no tracking errors. Definite evaluation of 
these prediction errors can be made only if the 
path of the target is prescribed. The simplest 
path which can be prescribed for this purpose 
is a circular one at constant velocity. Such a 
path is fairly realistic when considered in rela- 
tion to the difficulty of maneuvering a bomber 
and to actual records of the paths of hostile 
bombers over London during World War II. 

The position of a target flying in a circle at 
constant velocity, referred to the center of the 
circle, is expressed by the complex quantity 
Re^* where R is the radius of the circle and w 
is the angular rate. In terms of the velocity V 
and the transverse acceleration A, we have 
R = V^/A (0 = A/V. The predicted position is 
then at RY(io})e^^* where Y (iw) is the trans- 
mission function of the prediction circuit. The 
true future position of the target, however, is 
at R exp + tf)]. Hence, the prediction 

error, referred to axes fixed on the target and 
oriented respectively transverse to and in the 
direction of the present velocity, is 

e = R[Y(io)) — . 

As an illustration let us consider a case in 
which V = 150 yd per sec, A = 5 yd per sec^ and 
tf = 10. For the linear prediction circuit 

Yi(ioi) = 1.0409 + ?0.3296 
and for the quadratic prediction circuit 
7,(?co) = 0.9501 + i0.3610 

while 

= 0.9450 + iO.3272 . 

Hence, when the present position of the target 
is at 4500 + zO with respect to the center of the 
circle, the linear predicted point is at 4684 + 
H483, the quadratic predicted point is at 
4276 + H624 while the true future position is 
at 4252 + H472. These are shown in Figure 10. 
The prediction error vectors are 

ei = 432 + HI |ei| = 432 

e, = 24 + H52 lej = 154 . 


Referring to Figure 10 it may be observed 
that if the first-derivative component of the 
prediction were to be reduced by approximately 
10 per cent a nearly perfect hit would be ob- 
tained. This suggests the possibility of deter- 


2000 








QUADRATIC PREDICTED 
^ POSITION 

SECOND DERIVATIVE 

TRUE FUTURE 

POSITION 

(10 SEC) ^ 


LtAU VLCrOK 

LINEAR 

f^PREDICTED 





POSITION 






ivative 

:tor 

■ 1000 — 


• 

FIRST DEP 

/lead vec 



• 






. 










4500 YDS TO 
„ CENTER OF TURN , 

f PRESENT POSITION 


Figure 10. Vector diagram of linear and quadratic 
prediction for constant-velocity circular courses. 

mining empirical functions of the time of flight 
for the potentiometer factors and in 
order to improve the probability of kill. This 
would involve consideration of all of the 
sources of dispersion enumerated in the preced- 
ing section as well as a statistical study of tar- 
get paths. Such a determination has not been 
attempted. 

13.1.5 Physical Configuration of the 
Second-Derivative Circuit 

In this section we shall derive a physical con- 
figuration for the second-derivative circuit. In 
particular it illustrates the application of feed- 
back to the realization of weighting functions 
or impulsive admittances involving complex 
exponentials in general."^ It should be pointed 
out, however, that the application of feedback 
to the end in view is not restricted to purely 

^ Originally proposed by R. L. Dietzold. 


CIRCUIT FOR CLOSE SUPPORT PLOTTING BOARD 


131 


electronic circuits. An application involving 
the use of servomechanisms will be described 
in Section 13.4. 

The transmission function which concerns us 
here may be expressed in the partially factored 
form 


(p + 0.2087) (p + 1.5864) (p2 + 0.3049p + 0.0666) 
where the poles have been adjusted to cor- 
respond to r = 20 seconds and where a constant 
factor has been left out. 

The circuit is to be designed to work out of 
the amplifier in the first-derivative circuit of 
the M9 director. Since this much of the first- 
derivative circuit has a transmission function 
of the form p/(p + 0.24), the transmission 
function which we have to realize is Yi{p)/ 
Yf(v) where 

r P 

^ (p + 0.2087) (p+ 1.5864) 

and 

,, , , + 0.3049p + 0.0666 


The input network has four elements, 
whereas Ti (p) has only two parameters. Hence 
there are two degrees of freedom in the element 
values of this network. One degree of freedom 
must be reserved for the impedance level; the 
other permits some latitude in the relative 
values of the resistances and stiffnesses. 

The feedback network has four independent 
elements, whereas Yf{p) has three parameters. 
Hence there is only one degree of freedom in 
the element values of this network. This degree 
of freedom must be reserved for the impedance 
level. 

There is, however, one degree of freedom be- 
tween the impedance levels of the two net- 
works. This follows from the fact that the 
transmission function of the circuit is the ratio 
of the transmission functions of the individual 
networks. The scale factor for the transmission 
function of the circuit is readily determined 
from the fact that the transmission function 
must be approximately vRqCo at small values 
of p. 


The inversion of the factor corresponding to 
Yf{p) is in accordance with the fact that the 
transmission gain through a feedback amplifier 
is equal to the loss in the feedback network, 
provided the feedback is very large. To realize 
the transmission function Yi(p) /Yfip) it is 
therefore necessary only to realize the trans- 



Figure 11. Physical configuration of quadratic 
prediction circuit for modified M9 AA director. 


13 2 CIRCUIT FOR CLOSE SUPPORT 
PLOTTING BOARD 

In this application, position data smoothing 
with delay correction for constant rates of 
change in position was required. Assuming flat 
random noise in position data, and, arbitrarily, 
1-second smoothing time, the best transmission 
function for position data smoothing without 
delay correction is Vo (p) in the notation of 
Section 11.3. The best transmission function 
for the first-derivative circuit, if it were re- 
quired, is pPi (p) . Hence, the best transmission 
function for position data smoothing with full 
delay correction is 

Pi(p) = yo(p) + ^ pyi(p) . 

This corresponds to the weighting function 

Wl{t) = Wo{t) 

= 2(2 -3t) 0 < t < 1 . 


mission functions Fi(p) and F/(p) individu- 
ally. The corresponding networks are shown in 
Figure 11, with typical element values. 


The series expansion for F/(p) is, by (15) 
of Chapter 11, 


r,(p) = i -g + i; 




132 


ILLUSTRATIVE DESIGNS AND PERFORMANCE ANALYSIS 


The form of the rational approximation was 
chosen as 

1 + hip + 62^2 + 63^^ 

in order to obtain a loss characteristic which 
has an ultimate slope of 12 db per octave.® This 
requirement was also set as a precaution 
against noise due to granularity of the coordi- 
nate-conversion potentiometers. The coefficients 
are determined by 

hi = ai 

6. - = 0 

■“ l2^‘‘ ~ 120 ^ ^ 


whence 


Y(p) = 


1 . 

^ + 24^ 


. . 11 . 1 o , 7 

^+^^+12^+ 1440^ 


This may be expressed in the form Y (p) = 
Yi(p)/Yfip) where 


Yi(p) = 


1 + 0.1053p 


, 1 + 0.3530p + 0.0461 5p2 

1 + 0.4583p 

The circuit configuration is shown below in 
Figure 12. 



Figure 12. Physical configuration of data-smooth- 
ing circuit for close support plotting board. 


« This design also antedated the formulation of the 
n — m = r + 1 rule given in Section 12.2 according to 
which we should have taken Yi{p) = yi{p) + V 2 py^ip). 


3 CIRCUIT FOR GROUND-CONTROL 
BOMBING COMPUTER 

In this application, rate smoothing as well as 
position smoothing was required. In addition, 
delay correction in position, for constant rate 
of change, was to be available but optional, and 
the loss characteristic was to have an ultimate 
slope of 12 db per octave, or more. 

In accordance with the n — m = r Y 1 rule, 
the best transmission function for position data 
is Vi (p) , whereas that for rate is P 2/2 (p) • A num- 
ber of designs were made on this basis. How- 
ever, from the point of view of network econ- 
omy they were inferior to a design based on 
2/2 (P) for position data. The use of 2/2 (P) for 
position data is not consistent, theoretically, 
with the use of P 2 / 2 (p) for rate, but the practi- 
cal advantage outweighs the theoretical disad- 
vantage. 

The rational approximation used for 2/2 (P) 



R,C, = 0.6613 
= 0.2647 
RjCi = 0.5438 
RjC, = 0.5438 
RjCj = 0.5000 
■R 4 C 4 = 0.1360 
RsCs = 0.1360 


0.2153 

0.2153 


(FOR DELAY CORRECTION) 
(FOR FIRST DERIVATIVE) 
(FOR DELAY CORRECTION) 


Figure 13. Physical configuration of linear pre- 
dictipn circuit for ground-control bombing com- 
puter. 

is the one given in (6), Section 12.2. It may 
be expressed as 

Yii(p) . Yi2(p) 

vAp) 

where 

1 


2/2(p) = 


YMp) = 
Y,{p) = 

Yi2(p) = 


] yONFlDENTIAL^ 


1 + 0.2153p 

1 + 0.2847p + 0.03870p2 
1 + 0.1359p 

1 

1 + 0.1359p ■ 


CIRCUIT USING SERVOMECHANISMS 


133 


It may be noted that a redundant factor has 
been introduced, viz., 1 + 0.1359p, in order to 
secure a physically realizable Yf{v) . The coeffi- 
cient was chosen so that a resistance would not 
be required in the shunt branch of the feedback 
network. Referring to the circuit configura- 
tion in Figure 13, the transmission function of 
the input network is that of the feed- 

back network is Yf(p), and that of the output 
network at the top is Yi^{p) . 

The output impedance of the amplifier is re- 
duced nearly to zero by virtue of shunt feed- 
back.'^j Hence, the rate circuit, as shown in 
Figure 13, may be derived from the amplifier 
output through a simple additional network 
whose transmission function is pY i^ip) . Two 
rate outputs are provided so that the delay 
introduced in position may be corrected option- 
ally without disturbing scale factors. 

CIRCUIT USING SERVOMECHANISMS 

In the final report, October 25, 1945, to 
NDRC Division 7, on the research program car- 
ried on under Contract NDCrc-178, a list is 
given of a number of the more important prac- 
tical advantages for the use of a-c carrier in 
computing circuits. These advantages are: 

1. Permits operation at lower levels before 
running into trouble with thermal noise, contact 
potentials, drifts due to temperature ; 

2. Permits use of transformers for imped- 
ance matching, voltage transformations, cou- 
pling between balanced and unbalanced circuits ; 

3. Permits use of hybrid coils for voltage 
summations of moderate precision ; 

4. Eliminates the necessity for modulators in 
servo circuits using a-c motors ; 

5. Permits reduction in total power consump- 
tion, rectified power for amplifiers, and voltage 
regulation. 

However, the techniques of differentiation 
and of data smoothing with fixed networks in 
computing circuits which use d-c carrier, are 
not applicable to computing circuits which use 
a-c carrier. 

The circuit described here is an example of 
one of the techniques used in the T15-E1 experi- 
mental curved flight director.® In Figure 14 
servo motors^ are indicated by M, and genera- 

^ The technique of using servo motors for smoothing, 
as described above, is due chiefly to E. L. Norton. 


tors by G. The motors are two-phase induction 
motors with one phase winding of each ener- 
gized directly by the carrier source at constant 
amplitude. The generators are essentially two- 
phase induction motors also with one phase 
winding of each energized directly by the carrier 
source at constant amplitude. They deliver, at 



Figure 14. Electromechanical linear prediction 
circuit. 

the other phase windings, carrier voltage at 
amplitudes proportional to the angular velocities 
(9i and 6^ of the shafts. The potentiometers are 
energized by the carrier source at constant am- 
plitude. They deliver carrier voltage at ampli- 
tudes proportional to the angular positions 6^ 
and ^2 of the shafts from some reference posi- 
tions. The position data are represented by the 
modulation amplitude E. 

With amplifiers of sufficiently large voltage 
gain and power capacity, and motors of suffi- 
ciently large torque, the operational equations 
of the circuit are readily found by equating to 
zero the sum of the voltages applied to each 
amplifier. Thus 

+ («i + ^p)^2 = E 
pdi — (1 -f- a2p)S2 = 0 


a ^ ^ ^ p 

^ 1 -}- (ai -f a2)p + 

a _ P p 

1 + (ai + a2)p -\-0p^ 

The angular position therefore represents 
the smoothed position data while the angular 
position $2 represents the smoothed rate. 


Chapter 14 

VARIABLE AND NONLINEAR CIRCUITS 


T he past discussion has been more or less 
clearly directed at predictor systems hav- 
ing certain well-defined properties. For ex- 
ample, it has been tacitly assumed that the first 
part of the prediction system will consist of 
geometrical manipulations transforming the 
raw input data into other quantities, such as 
the components of velocity in Cartesian or in- 
trinsic coordinates, which we have some physi- 
cal reason to believe should be approximately 
constant for extended periods.^ These quanti- 
ties, then, are isolated explicitly in the circuit 
and are the actual effective inputs of the data- 
smoothing networks. The data-smoothing net- 
works themselves are, of course, definitely 
assumed to be linear and invariable. 

This is obviously a straightforward attack 
but it does not necessarily exhaust all possibili- 
ties. For example, advantages may be gained 
by using data-smoothing networks which are 
nonlinear or which vary with time or target 
position. It may also be possible to smooth the 
input data according to some geometric as- 
sumption, such as straight line flight, without 
the necessity of isolating geometrical parame- 
ters explicitly. 

This chapter attempts to illustrate these pos- 
sibilities by some rather scattered examples. 
Data-smoothing networks which vary with time 
seem to give improved performance over fixed 
networks, and have been studied with some 
care. Several examples are given at the end of 
the chapter. None of the other lines, however, 
has been explored at all thoroughly. The ex- 
amples of data-smoothing networks variable 
with time are, in a sense, illustrations of non- 
linearity also, since they all operate on the 
assumption that the cycle of the network's 
variation with time begins anew at each 
marked change in course. Since a change in 
course is exactly like a tracking error, except 
that it is much larger, this resetting requires 
a nonlinear control circuit which will respond 
to large amplitude effects but not to small ones. 


^ This is true ideally even in the Wiener system since 
Wiener assumes that transformations will be made to 
some suitable coordinate system, preferably the intrin- 
sic, before the statistical prediction method is applied. 


This, however, is evidently a very mild sort of 
nonlinearity. More thoroughgoing nonlineari- 
ties have not been studied. There seems to be 
no a priori reason for supposing that they 
would appreciably improve the performance 
of data-smoothing networks. 

The first part of the chapter gives examples 
of data-smoothing schemes which do not re- 
quire the isolation of geometrical parameters. 
They are based on degenerative feedback cir- 
cuits which satisfy the requisite formal rela- 
tions but which might, in some cases, be un- 
stable in practice. This portion of the material 
is included primarily for its possible sugges- 
tive value rather than for its concrete practical 
usefulness. 

1 THE PROTOTYPE FEEDBACK 
CIRCUIT 

The diversity of particular circuits can be 
given a certain unity by regarding them all as 
modifications of the feedback smoothing cir- 
cuit shown originally in Figure 2 of Chapter 
10. In accordance with the discussion of that 
figure it will be convenient to suppose that the 
resistive feedback path is introduced to limit 
the gain of the amplifier proper, so that the 
structure reduces to an amplifier with high but 
finite gain and a pure capacity feedback. The 
circuit has a net loop gain, and is consequently 
degenerative, at any moderately high frequency. 
For our present purposes, it is convenient to 
recall the general property of degenerative 
feedback amplifiers, that they tend to suppress 
any given frequency by the amount of the de- 
generative feedback for that frequency. This 
suppression obtains not only at the amplifier 
output but at many other points in the circuit 
as well. For example, it holds at the amplifier 
input if we combine the original applied volt- 
age with the voltage contributed by the feed- 
back*’ circuit.i'^s Thus, except for the absolute 


*» This follows immediately from the fact that, since 
the characteristics of the amplifier proper are not 
changed by the addition of the feedback path, the 
output voltage is always a fixed multiple of the net 
input voltage. 


134 


^NFIDENTIA 


SIMULTANEOUS SMOOTHING IN THREE COORDINATES 


135 


signal level, it is not necessary to transmit 
through the amplifier of Figure 2 of Chapter 
10 in order to produce the smoothing effect. It 
would be sufficient to hang the input circuit of 
the amplifier, as a two-terminal impedance, 
across the circuit. 

142 SIMULTANEOUS SMOOTHING IN 
THREE COORDINATES 

The property of degenerative feedback cir- 
cuits which has just been described is con- 
veniently illustrated by a three-dimensional ex- 
tension of the original smoothing circuit of 
Figure 2 of Chapter 10. The three-dimensional 
circuit is shown in Figure 1. The three input 
voltages are the quantities D, DE, and DA cos 



Figure 1. Feedback smoothing in three coordinates. 


E, where D, E, and A are, respectively, slant 
range, elevation, and azimuth. The three volt- 
ages will be recognized as the three components 
of the target motion in a tilted and rotating 
rectangular coordinate system. One axis of the 
tilted system is directed along the instan- 


taneous line of sight to the target and the other 
two are perpendicular to this one in the ver- 
tical and horizontal planes respectively.® It is 
assumed that these input rates represent target 
motion in a straight line, plus the usual track- 
ing errors. The object of the smoothing system 
is to provide shunt impedances which will tend 
to suppress the tracking errors by feedback 
action, according to the principles described in 
the preceding section, without disturbing the 
portions of the input voltages corresponding to 
the assumed straight line path. 

We can simplify the analysis by restricting 
our attention to the special case of two-dimen- 
sional motion which occurs when the target 
course lies in a vertical plane passing directly 
through the antiaircraft position. This is illus- 
trated in Figure 2. In this case the component 
DA cos E is evidently zero. If we represent 
the voltage at the other two terminals, includ- 
ing both the original applied voltages and the 
voltages fed back through the circuit, by and 
Fj, the voltages coming out of the coordinate 
converter on the right-hand side in Figure 2 
are 

Vx = Fi cos E — V 2 sin E 

Vy = F 2 cos E A Fi sin F . (1) 

These voltages are differentiated, passed 
through a second coordinate converter, and fed 
back so that the output voltages must satisfy 
the equations 

Fi = D — ijl{vx cos E + Vy sin E) 

F 2 = DE — ix{vy cos E — Vx sin E) . (2) 

In order to exhibit the smoothing action of 
the circuit let us denote the observed velocity 
components, referred to the upright and fixed 


« This is the coordinate system which was used in the 
experimental T15 director. A complete prediction cir- 
cuit can be obtained by using the three voltages de- 
scribed here as inputs to the lead servos in the T15 
system. In the actual T15 system, rates in the tilted 
and rotating coordinate system were obtained by the 
so-called ^‘memory point” method. The voltages D, DE, 
etc., required with the present method, might be ob- 
tained with the help of tachometers attached to the 
tracking shafts to measure the instantaneous values of 
b, E, and A. An equivalent to the variable smoothing 
of the memory point method can be obtained by making 
the gains in the feedback paths in Figure 1 variable 
according to the principles described in a later section. 


^NFIDENTIAL ^ 


136 


VARIABLE AND NONLINEAR CIRCUITS 


rectangular coordinate system, by and Uyy 
so that 

Ux = D cos E — DE sin E 


tially the tracking errors only. Since tracking 
errors are always small, very high percentage 
errors in the system can be tolerated.^ 


Uy = DE cos E D sin E . (3) 

Substituting (2) and (3) into (1), we get 

Vx = Ux — flVx 
Vy = Uy - fli'y 


fJ-Vy Vy Uy . 

These show clearly that Vx and Vy are smoothed 
values of Ux and Uy, respectively. If is constant 
the smoothing is of fixed exponential type. If fi 
is proportional to the time up to some maxi- 
mum value, the smoothing is of the variable 
type described in Sections 14.6 and 14.7. 

To complete the discussion of the circuit we 
observe that by (1) 

Vi = Vx cos E + Vy sin E 

Vi = Vy cos E — Vx sin E . 

These show that and are the smoothed 
rate components referred to the tilted and 
rotating rectangular coordinate system. The 
fact that the orientation of this coordinate sys- 
tem, which depends upon the observed angular 
height E, is not smoothed makes no difference 
to the computation of the leads because this 
computation is made instantaneously in the 
same coordinate system to which the smoothed 
rate components are instantaneously referred. 

The analysis in the general case including 
all three coordinates is of the same nature. 
Since the rate components in fixed rectangular 
coordinates appear in the middle of the feed- 
back path, it is perhaps not fair to regard the 
circuit as an illustration of a data-smoothing 
device which does not rely upon the explicit 
isolation of the geometrical parameters of the 
assumed target path. It should be pointed out, 
however, that in comparison with a straight- 
forward geometrical solution in which velocity 
components in fixed coordinates are first isolated 
explicity, then smoothed, and then used to form 
the basis of prediction, the circuit in Figure 1 
has the advantage that most of the components 
can be built with very low precision. What is 
transmitted around the feedback loop is essen- 




143 SMOOTHING NETWORKS VARIABLE 
WITH TARGET POSITION 

It was mentioned earlier that changing the 
data-smoothing network with the target coor- 
dinates represented one way in which the re- 
sults obtained from fixed networks could be 

4 An exception to this statement must be made for 
errors in the coordinate converters which fluctuate 
rapidly with target position. 


»NFIDENTIAl 



SMOOTHING NETWORKS VARIABLE WITH TARGET POSITION 


137 


generalized. In a sense, the coordinate conver- 
sions of Figure 1 are illustrations of these 
possibilities. A better illustration, however, is 
provided by the circuit of Figure 3. The struc- 



Figure 3. Feedback smoothing with smoothing 
variable with position coordinates. 


ture is intended to give smooth slant range 
rate from slant range data, under the assump- 
tion of unaccelerated straight line target 
motion. 

The relation between input and output in 
Figure 3 is readily seen to be ® 

or 


M (DV) + 7 = 
at 


dt 


(4) 


where /a is the amplifier gain, D is slant range, 
and V = dD/dt is slant range rate. 

The principle of the circuit depends upon the 
fact that under the assumed target motion the 
square of the slant range, should be a 
quadratic function of time, so that [D (dD/dt)] 
should be a linear function of time and (d/dt) 
\_D (dD/dt)] should be a constant. This last is 
the quantity which is fed back in Figure 3. 
If it actually is a constant, it has no further 
influence on the calculation, since the forward 
circuit includes a differentiator, and the opera- 
tion of the circuit is the same as though no 
feedback term were present. This can be verified 
by setting D = D^ = y/a + 2ht + corre- 
sponding to ideal straight line flight, in equa- 
tion (4) . It is readily seen that the equation is 
satisfied by 

7 = ^ _ dP^ 

's/ a -|- 2ibt -|- ct^ dt * 

the first or feedback term being zero. 


® The condensers in Figure 3 symbolize differentia- 
tion. 


If D does not correspond exactly to straight 
line Alight, either because of tracking errors 
or actual target maneuvers, on the other hand, 
the feedback voltage is no longer constant. In 
this case transmission around the loop can 
exist and the degenerative feedback action 
produces smoothing in both the input and the 
output voltage. In calculating the exact effect 
we must take account of the fact that the feed- 
back voltage depends upon the D potentiometer 
in the feedback circuit as well as upon the out- 
put voltage V. Since the D potentiometer set- 
ting must include the errors in the input data, 
this means that the output voltage is not per- 
fectly smoothed, even with unlimited gain 
around the loop. The percentage error in the 
output rate tends in the limit to approximate 
the percentage error in D itself. For practical 
purposes, however, this is a very satisfactory 
result, since in the absence of smoothing per- 
centage errors in rates are usually many times 
those of the corresponding coordinates. 

It is apparent that it should be possible to 
construct many circuits of this general type 
from the differential equations of the trajec- 
tory. A second example is furnished by Figure 
4. The operation of the circuit is essentially 



Figure 4. Another example of feedback smooth- 
ing with smoothing variable with position coordi- 
nates. 

similar to that of Figure 3. It depends upon 
the fact that in unaccelerated straight line 
motion the quantity D-A cos^ £* is a constant. 
Instead of multiplying by D^ and cos^ £■ at a 
single point in the feedback loop, however, 
separate multiplications by D and cos E are 
introduced in the forward and feedback cir- 
cuits. This permits the output to appear as a 
smoothed value of the quantity DA cos E, 




138 


VARIABLE AND NONLINEAR CIRCUITS 


which will be recalled as one of the primary 
quantities in the circuit of Figure 1. 

14.4 networks variable with time 

In addition to making the parameters of the 
data-smoothing network vary as functions of 
the coordinates of target position we may also 
make them variable as functions of time. The 
advantage of variation with time can be under- 
stood by going back to the discussion of the 
analytic arc assumption and its consequences 
for fixed data-smoothing networks, as given in 
Chapters 9, 10, and 11. It will be recalled that 
for any given settling time there was an opti- 
mum choice of the network’s weighting func- 
tion. The choice of the settling time itself, how- 
ever, was always a compromise. On the one 
hand, making the settling time too short led 
to too little smoothing, so that the dispersion 
in the resulting fire became excessive. On the 
other hand, too long a settling time meant that 
data from previous unrelated segments were 
retained in the smoothing circuit during too 
large a proportion of an average individual seg- 
ment of the target path, leaving too small a 
residue of the average segment as useful firing 
time. 

It is evident that it is theoretically possible 
to escape the consequences of this compromise 
by resorting to variable structures. We need 
merely assume that the network always has a 
weighting function appropriate for a settling 
time equal to the time since the last change in 
course. This would give a small amount of 
smoothing shortly after a change in course, 
with more smoothing and consequently greater 
accuracy later on. No firing time, however, is 
sacrificed waiting for the network to settle. 

In order to exploit these possibilities we 
must, of course, be able to design networks to 
give at least approximately the right sequence 
of weighting function. It is also necessary to 
provide some sort of auxiliary controlling 
mechanism which will sense changes in target 
course and return the variable circuits in the 
smoothing network proper to their initial posi- 
tions. These are both difficult problems which 
have been incompletely explored. Some elemen- 
tary solutions, based principally upon modifica- 
tions of the degenerative feedback smoothing 


circuit of Figure 2, of Chapter 10, are, how- 
ever, given later in the chapter. As a prelimi- 
nary, the next section gives a formal extension 
of the general polynomial expansion method of 
Chapter 11 to the variable case. 


1^5 GENERAL POLYNOMIAL SOLUTION 
FOR VARIABLE NETWORKS 

The extension of the general method of 
Chapter 11 to the variable case requires two 
modifications. 

1. The lower limit of the integral to be 
minimized is now taken as zero, in anticipation 
of the possibility of discriminating between rele- 
vant and irrelevant data on the basis of time of 
arrival. 

2. The weighting function may now depend 
more generally upon the variable of integration 
and the upper limit of integration. 

With these modifications there is no longer 
any advantage in conducting the analysis in 
terms of the age variable t. To deal directly 
with the minimization of the integral 

£ [£(X) - £(X)]2 W"„(/,X) rfx , (5) 

let 

E{\) = Fo + Fi • GiftX) + . . . + Fn • (6) 

Where Gm{tyX) is an mth degree polynomial in 
A. Also, let 



t 

• Wo(^,X) dX = 0 if / 7*^ m 

= -r if / = m 
km 

{Go = Ij ko = 1) . 

Then (5) is a minimum with respect to the 
F„/s in (6) if 

v„{t) =£ E{\) ■ Tf„(;.x) dx (7) 

where 

W..(/,X) = kmGm{t,\) • IFo(f,X) . (8) 

The possibility of physically realizing the 
Vm(t) depends upon the possibility of realizing 
networks with impulsive admittances 
in the sense that WmitjX) is the response of a 



NETWORKS WITH A LIMITED RANGE OF VARIATION 


139 


network, at time t, to a unit impulse applied at 
time A, where 0 < A < ^. Taking this possibility 
for granted, the predicted value E(t + tf) is, 
according to (6), a variable linear combination 
of the Vm(t), viz., 

E{t -f" tf) = To(0 + “h if) • ^i(0 + • • * 

+ Gnit,t -j- tf) • Vn{t)‘ (9) 

It is clear that all of the as well as 

all of the Gm(t,k) for m = 1, 2, . . . are deter- 
mined by Wo(t,x). The latter is determined as 
the best weighting function for position data 
smoothing, depending upon the characteristics 
of the noise associated with the position data. 
The general methods of determining the best 
weighting function with fixed smoothing time, 
described in Chapter 10, may be used to deter- 
mine the best weighting function with variable 
smoothing time. 

Under the assumption that the spectrum of 
the noise associated with the signal S{t) has a 
uniform slope of 6k db per octave, we may take 
over from Section 11.3 the result that the best 
weighting function is 

0 < \ < t . 

The response of the network is then 

Vit) = jT S{\) ■ «)*(/, X) dx . (11) 

SPECIAL CASES 

It will be illuminating to consider a few 
special cases of (11). 

For A; = 0, we have 


Multiplying through by P and differentiating 
twice we get 

+ tv + V = s 

6 

which may be written in the form 

(la + OGl, + 

This suggests the network shown in Figure 6.« 


NETWORKS WITH A LIMITED 
RANGE OF VARIATION 

By generalizing the above results in various 
ways a large number of other examples of 
variable smoothing networks can be constructed. 
Since unlimited variation in the smoothing 
time is not practically possible, or perhaps even 
tactically optimal, however, it is desirable in 
discussing any further examples to include also 
the possibility that the range of variation in 
the network may be restricted. For any posi- 
tive integral value of /c in (11) the differential 
equation for V (t) is of the type which may be 
reduced by the transformation t = to sl linear 
differential equation with constant coefficients.^ 
In general, this facilitates the determination of 
what happens to the weighting function 
Wk{t,\) when t>T it the variability of the 
network is stopped at time T, In the case of the 
first-order equation (13), however, it is just 
as easy to deal directly in terms of the natural 
time. 

A more general form for (13), which readily 
yields the effects of a sudden or gradual stop- 
page of the variability of the network, is 


Vit) 


r 

t Jo 


S{\) dX . 


( 12 ) 


m + V{t) = S(t) 


Multiplying through by t and differentiating This corresponds to the response 
we get 


tVit) -h Vit) = Sit) . (13) 

This suggests the circuit shown in Figure 5.^ 
For k = 1, we have 


Vit) 


A r 

<f>(f) Jo 


S(\) ■ dX 


Vit) 


= - r 

Jo 


whence the weighting function is 

^(X) 


w(t,X) = 


Six) ■ X(< -X) dX 


<t>it) 


(14) 


(15) 


« Due to B. T. Weber. 

*^See Section A.ll for a more general transforma- 
tion. 




CONFIDENTIAL 



< This circuit is due to S. Darlington. 


140 


VARIABLE AND NONLINEAR CIRCUITS 


The general relation (14) may be realized 
with the network of Figure 5, by varying the 
resistance in accordance with 


R=m 


t > 0 . 


Cct>{t) 

However, a more practical circuit results from 
the introduction of variable potentiometers* in 
both the capacity and resistance paths of the 




Figure 6. Time-variable smoothing circuit giv- 
ing parabolic weighting function. 

As an example of (14) we may take 
4>(0 = < 0 < t < T 
= Te''-’’*’’ t >T . 

Then 

= 7 t> T . 

Hence, in Figure 7, if RC = T 


fcit) = 


T 
= 1 


fR(t) =0 0 <t < T 
= 1 t> T . 


This example obviously calls for a linear poten- 
tiometer in the condenser path and a switch in 
the resistance path. The weighting function ob- 
tained is, by (15), 




1 


0 <\ < t < T 


= _ p-{t-T)/T 

T 


0 < \ < T < t 


= j 0 < T < \ <t 


Figure 5. Time-variable smoothing circuit giving 
uniform weighting function. 

original feedback smoothing circuit of Figure 
2, Chapter 10. This is shown in Figure It 
may be noted that the feedback circuit is also 
applicable to the two cases discussed in the 
preceding section. It has the advantage for 
these applications that it does not require the 
zero-impedance generators and infinite-imped- 
ance loads of Figures 5 and 6. 



Figure 7. Limiteii range time-variable feedback 
smoothing circuit. 

This is illustrated in Figure 8 for 10, ^ = 5, 

10 , 20 . 


0.2 


0.1 


t = 5 


t = IO 


t=20 


T=I0 


15 20 

Figure 8. First example of weighting function 
produced by circuit of Figure 7. 


A second example is furnished by taking 
0(0 =0 0 < t < T 

= t > T . 


Then 


^ In some cases a variable potentiometer may turn 
out to be a switch. 

1 This circuit is due to S. Darlington. 


0(0 

0(0 


k 

T 

k 


0 < t < T 
t> T • 


CONFIDEOT 


OTHER EXAMPLES 


141 


Hence in Figure 7, if RC = T/k, 


The weighting function obtained is, by (15) , 


fed) 


i fnit) = 1 - r 0 < ' < r 


1 _± 

2T 

0 < X < / < T 


= 1 = 1 t> T . 

The first example is a special case of this one. 
The weighting function obtained is, by (15), 

h\k-l 

w(tX) = — ^ 0 < \ < t < T 


rpk 


e -k(t-T)/T 0 < \ < T <t 



Figure 9. Second example of weighting function 
produced by circuit of Figure 7. 

A third example is furnished by taking 
t 


<f>(t) = 


2 - 


t 


0 <t < T 


Then 


= Te 2(/-r)/r t > T . 


«<'< 




2T 




_ ^-m-T)/T 0 < X < T < < 


= Y 0 < T < X < < . 

This is illustrated in Figure 10 for T = 10, 
t = 5, 10, 20. 


= -f e 0 < T < X < / . 


This is illustrated in Figure 9 for k = 3/2, 
T = 10, f = 5, 10, 20. 



Figure 10. Third example of weighting function 
produced by circuit of Figure 7. 

A fourth example is furnished by taking 


^{t) = - 1 / > 0 . 


Then 


0(0 1 

Hence, in Figure 7, if RC = 1/k, 

fed) = /«(/) = 1 - t> 0. 

The weighting function obtained is, by (15), 
k 


wdX) = 


1 


^-k(t-\) 0 < X < < . 


For any value of t this weighting function is 
exponential in A. 


= 2 


Hence, in Figure 7, if RC = T/2, 

fc(t) = /««) = 4 0 < ' < '^ 

= 1 = 1 < > T. 

^CONFIDENTIAl^ 


“8 OTHER EXAMPLES 

Because there has been no demand for varia- 
ble networks in the field of communications, 
the technique of designing practical variable 
networks is' in a very rudimentary stage com- 
pared to that of designing fixed networks. In 
the remainder of this chapter we shall describe 


142 


VARIABLE AND NONLINEAR CIRCUITS 


some of the circuits which have been developed 
for specific practical applications. 

A memory point method of obtaining 
smoothed rates, based upon (12), is illustrated 
below. If S(t), the quantity to be smoothed, 
represents the time derivative E (t) of the posi- 
tion data E {t) , then the average rate is given 
by 

y(0 = . (16) 

Under the assumption that the position data, 
aside from tracking errors, is a linear function 
of time, the average rate is also the smoothed 
rate. If the position data is represented by the 
angular displacement of a shaft in the com- 
puter, the quantity E {0) is readily fixed by 
providing a second shaft which is coupled to 
the first shaft until ^ = 0 when the coupling is 
broken. Potentiometers mounted on the shafts 
are energized by a voltage varying as a func- 
tion of time in the manner indicated in Figure 
11. The manner in which the smoothed rate is 
obtained is clear. 



Figure 11. Memory point method of obtaining 
smoothed rate. 


The memory point method of obtaining 
smoothed rates is used in the T15 antiaircraft 
director.^ In this application, however, it is 
somewhat more complicated than in the simple 
illustration described above. This is due to the 
fact that the position data and the memory 
point are in the polar coordinate system, 
whereas the rate components are referred to 
a tilted and rotating rectangular coordinate 
system which is determined by the instanta- 
neous line of sight. 


Figure 12, shows a way of securing variable 
smoothing in a purely electrical circuit. Except 
for the fact that the division of the current 
through the condensers is varied discontinu- 



Figure 12. Specific limited range time-variable 
feedback smoothing circuit. 

ously instead of continuously, this circuit cor- 
responds to the first or the second example dis- 
cussed in Section 14.7. 

Figure 13 shows the variable smoothing cir- 
cuit ^ for smoothing first derivatives in the 
M9A1-E1 antiaircraft director.® This circuit 

R 



Figure 13. Another specific limited range time- 
variable feedback smoothing circuit. 


corresponds approximately to the second exam- 
ple of the differential equation (14) given 
above. The variable element is a thermistor 
which is heated up to a high temperature, prac- 
tically instantaneously, by the heater, and then 


^ This circuit is due to S. Darlington. 
1 Developed by R. F. Wick. 


OTHER EXAMPLES 


143 


allowed to cool off naturally. By choosing the 
electrical and thermal constants in the circuit 
correctly the resulting smoothing can be made 
to approximate that obtained in a memory 
point circuit. 

As noted earlier, all these variable circuits 
require some auxiliary control means to reset 
the variable circuits to zero whenever a new 
target is engaged or the current target makes 
a sudden change in course. In the T15 memory 
point system this function was performed by an 
operator. The operator was aided by a series of 
meters which compared the instantaneous 
memory point rates with average rates set in 
some time previously by hand. The visual in- 
dication of a change in course, calling for the 
selection of a new memory point, was a rela- 
tively large, smoothly and decisively varying 
deflection on the meters. In contrast, normal 
tracking errors appeared as relatively small 
random fluctuations of the needles. The circuits 
of Figures 7 and 12, which were intended for 
bombsight applications, were also under the 
control of an operator, who was supposed to 
start the mechanism at the beginning of each 
bombing run. 

Two control methods were used for the cir- 
cuit of Figure 13. In one, large changes in rate, 
corresponding to probable changes in target 


course, were distinguished by comparing the 
instantaneous value of the target rate, as ob- 
tained directly from a differentiator, with the 
smoothed value obtained at the output of the 
smoothing circuit. In the other method, equiva- 
lent information was obtained by again differ- 
entiating the instantaneous value of the target 
rate, making a second derivative of the target 
coordinate. In either case this rate difference 
or second derivative information was used to 
control a gas tube, which went off, supplying 
heating current to the variable thermistor, 
whenever the voltage applied to it exceeded a 
certain threshold. This threshold evidently 
marks the minimum change in course for which 
the variable network will be reset. In order to 
permit the use of a low threshold, without 
making the circuit unduly liable to false opera- 
tion because of the effect of tracking errors, 
the gas tube input voltage was first transmitted 
through a low-pass filter which suppressed 
most of the energy due to tracking errors. A 
considerable amount of work was done on the 
proportioning of this filter to provide the best 
protection against false operation with a low 
threshold and with minimum delay in resetting 
in case a change of course actually does occur, 
but the problem remains an interesting subject 
for research. 


(CONFIDEXTIAL 



APPENDIX A 


NETWORK THEORY 


T his appendix gives a summary of linear 
network theory which is pertinent to the 
analysis and design of data-smoothing and 
prediction circuits. It is incomplete in many 
respects and should therefore be supplemented 
by reference to established textbooks on the 
subject. However, it contains some results 
which are new. 

The present summary will be concerned 
mainly with fixed linear networks. Variable 
linear networks will be considered briefly in 
the last section. 


^ 1 IMPULSIVE ADMITTANCE 

A fixed linear transmission network is one in 
which the response V (t) is related to the im- 
pressed signal E(t) by a linear differential 
equation of the form 


t = A is conventionally denoted by the singular 
function So{t — A) where 

6o(r) =0 if r 7 ^ 0 
^ 5o(T)dT =0 if f < 0 
= 1 if f > 0 

The response of a fixed network to an im- 
pulse or any form of signal is independent of 
the time at which the signal is applied, provided 
it is expressed as a function of the time relative 
to the application of the signal. Let W (t) be 
the response to the signal Soit). This is called 
the “impulsive admittance’' of the network. 
Physically, it must be identically zero for nega- 
tive values of t. For an impulse applied at t = A 
the response will therefore be W {t — A) , which 
is identically zero for t < \. 

A physical signal Eit) such as the one shown 
in Figure 2 may be resolved into an infinite 


bn 


d{tY 


+ * • • + ^oF 


d-E ^ d^-^E ^ ^ 

with constant coefficients. It is well-known that 
the solutions of such a differential equation 
obey the “superposition principle.” This makes 
it possible to formulate the response of the net- 
work to any signal, in terms of its response to 
certain standard signals. 

A convenient standard signal for analytical 
purposes is the “unit impulse.” It may be re- 
garded as the limit of the rectangular pulse 
shown in Figure 1 as the duration of the pulse 


1 

Figure 1. Rectangular pulse signal. 

is decreased indefinitely while the amplitude is 
increased in such a way that the area under 
the pulse is always unity. The limiting function 
thus defined does not exist in a strict mathe- 
matical sense. However, it is very convenient 
for analytical purposes, and seldom leads to 
difficulties, to proceed as though the limiting 
function did exist. An impulse occurring at 


X 

X =t 

Figure 2. Derivation of superposition theorem. 

succession of elementary impulses. The strength 
of the typical elementary impulsive component, 
such as the one shown in Figure 2 as occurring 
at time A, is E(x)dk. Its contribution to the 
response at time t is E{\)-W{t — A)dA. Hence 
the contribution of all the elementary impulsive 
components of the signal, to the response at 
time t, is given by the formula 

V(t) = EW ■ W(t - X)d\ (2) 

This is one form of the “superposition theo- 
rem” for fixed linear networks. 

Before discussing the reasons for the limits 
of integration indicated in (2), it will be help- 
ful to consider a graphical interpretation other 
than the one used in deriving the integral. Let 
W {t) be of the form shown in Figure 3, and let 
E(x) be of the form shown in Figure 4. To 
determine the response V (t) at a given value 
of t, the curve in Figure 3 is turned over from 



\CONFIDENTIAI 


145 


146 


APPENDIX A 


right to left and placed over the curve in Fig- 
ure 4 so that its right-hand edge is at A. = t. The 
product of the two curves gives a third curve 
(not shown), which is identically zero for all 
\> t The area under the third curve is the re- 



0 1 


% 

Figure 3. An illustrative impulsive admittance. 

sponse V (t) at the given value of t. For pro- 
gressively larger values of t, the curve repre- 
senting W (t — \) in Figure 4 is simply slid to 
the right with respect to the curve represent- 
ing EM. 


signal, is also expressed by the point of view 
that a fixed network cannot make any physical 
distinction between having no applied signal 
and having an applied signal which happens to 
be of zero amplitude. 

Another shortcoming of the form (3) or, for 
that matter, of the form (2) if we set t as the 
upper limit of integration, comes from the con- 
sideration of impulsive admittances of such a 
nature that Wit — X) has certain kinds of sin- 
gularities Sit X = t. For example, the case for 
direct transmission, expressed in the form 



F(X) • 8o{t - X)d\ 


is ambiguous because the singularity in the 
integrand occurs exactly at one end of the 
range of integration. However, the form 



Figure 4. Graphical interpretation of superposi- 
tion theorem. 


Since a physical signal must certainly be 
identically zero up to some definite time, or 
since it must certainly have been applied to the 
network at some definite time, that time could 
be taken arbitrarily as zero and (2) could be 
written in the form 


V{i) = - Wit - \)d\ (3) 

In this form, however, since 



W{t ~ \)dX 



is in general a function of t, the response could 
not be interpreted as a weighted average of the 
signal. On the other hand, since 



W(t -\)dX = 



WiT)dT 


is independent of t, the response may be inter- 
preted as a weighted average of the signal, if 


V(t) = E{\) ■ 6o(< - \)d\ 

leads, without ambiguity, to the result 
V (t) = E(t) . This example is not trivial. Every 
network which transmits infinite frequency 
must have an impulsive admittance of such a 
nature that W (t — X) contains a singularity of 
the form ~ A) . Any attempt to rule out such 
a singularity on the ground that physical net- 
works cannot in fact transmit infinite fre- 
quency, complicates the analysis and design of 
networks unduly. If a network is capable of, 
or is expected to transmit frequencies at the 
top of the range of interest or importance, it is 
simpler to assume that the network is capable 
of, or is expected to transmit all frequencies 
above that range. 

One other advantage of taking the limits of 
integration as indicated in (2) may be called 
to attention. Keeping in mind that E(x) is 
identically zero for all values of X below some 
definite though perhaps unknown value, and 
that IF(^ — A) is identically zero for all values 
of A > t, it is clear that (2) may be integrated 
partially any number of times without incur- 
ring the burden of carrying a string of terms 
outside of the integral. After one partial inte- 
gration we have 

V{t) = E'{X) • Ait - \)dx (4) 




WiT)dT = 1 


where 


The necessity of taking the lower limit in (2) 
as — C50, in order to permit the interpretation 
of the response as a weighted average of the 


A(t) 




WMdr 


(5) 



Since E'ix) is identically zero for all values of 
A in which EM is identically zero, and since 


FIDENTIAL 


APPENDIX A 


147 


A(^ — A) is identically zero for all values of 
A > a second partial integration may be per- 
formed with no more formal complication than 
the first partial integration. The fact of the 
matter is that the terms which ordinarily arise 
in partial integrations, outside of the integral, 
are here carried under the integral by singulari- 
ties of the integrand. 

The superposition theorem in the form (4) 
may be derived directly in a manner similar to 
the derivation of (2). A(t — X) is the response 
of the network to a Heaviside unit step func- 
tion H(t — A) applied at t = A, where 

H{t— X) = 0 when f < X 
= 1 when t > X . 

The signal is resolved into an infinite succes- 
sion of elementary step functions of amplitude 
E'(x)dx wherever E (x) is continuous, and 
finite step functions of amplitude dE (a) wher- 
ever E(x) has a finite discontinuity. The con- 
tribution of each elementary step function to the 
response at time t is E' (x) •Ait — x)dx, that 
of each finite step function is Ait — x) • dEix). 
Hence, the response is given formally by (4) 
with the understanding that E'ix) dx is to be 
interpreted as dE ix) wherever E ix) is discon- 
tinuous.^ 

The response A (0 of the network to a 
Heaviside unit step function Hit) applied at 
t = 0 is called the “indicial admittance’’ of the 
network. It is more familiar, in the field of 
linear transmission theory, than the impulsive 
admittance to which it is related by (5), but in 
this monograph preference is given to the use 
of the impulsive admittance. In the theory of 
linear differential equations the impulsive ad- 
mittance is known as a Green’s function. 

It is often convenient to express the response 
so that the variable of integration represents 
the age of the elementary components of the 
signal. Introducing the age variable 

T = t — X (6) 

into (2), we have 

V{t) = JJfS-t) ■ W{r)dT. (7) 

=■ Formula (4) may be written in the Stieltjes form 

V(t)= dE(x). 

Alternatively, we may take the point of view that 
E'(A) contains impulsive singularities whereVer Eix) 
is discontinuous. This point of view is generalized in 
Appendix B. 


In this form it is clear that the weighting of 
signal components is on the basis of age only. 
A fixed network may be said to have a memory 
which is a function only of the age of past 
events. 

In the preliminary stages of designing a 
smoothing network, the weighting function 
W ir) is generally prescribed to be identically 
zero when t> T say, as well as when t < 0. 
This does not violate the conditions of physical 
realizability^ However, such a weighting func- 
tion cannot be obtained exactly with a network 
of a finite number of discrete impedance ele- 
ments. A finite network invariably yields a 
weighting function with a “tail” which extends 
to infinity. 

A 2 TRANSMISSION FUNCTION 

Theoretically, the impulsive admittance of a 
prescribed network may be determined directly 
from the differential equations of the network 
in a perfectly straightforward manner. Prac- 
tically, however, it is very difficult to do so if 
the network has more than two meshes. Fur- 
thermore, the technical problem of designing 
a network directly from a prescribed impulsive 
admittance is even more difficult, particularly 
if the impulsive admittance is not exactly re- 
alizable. 

These difficulties may be avoided by recourse 
to the highly developed methods of network 
analysis and synthesis used in the field of com- 
munication circuits. These methods are based 
upon the steady-state properties of networks. 

If a signal consisting of the single sinusoid 
cos wt is applied to an invariable or fixed 
linear transmission network, the steady-state re- 
sponse*" will also be a single sinusoid of the 
same frequency. The amplitude and phase of 
the response, relative to the signal, will in 
general depend upon the frequency. The re- 
sponse may be regarded as the resultant of an 
“inphase component” proportional to cos wt, 
and a “quadrature component” proportional to 
sin tot, with amplitude coefficients which are 
functions of the frequency. Furthermore, since 
the signal is an even function of the frequency, 
the response should also be an even function 
of the frequency.® Hence, the response will 

^ This is the response apart from transient compo- 
nents, assuming that the latter vanish exponentially 
with time after the signal is impressed. 

® The signal is also an even function of the time but 
this is due only to the particular choice of origin which 
is arbitrary. 


148 


APPENDIX A 


be of the form Giio^) cos U — o)H(o)~) sin wty 
where G and H are even real functions of fre- 
quency. 

By a suitable shift of the origin of time it 
follows that if the impressed signal is sin wt, 
the steady-state response will be of the form 
G sin (Jit + o)H ((o^) cos cot. 

These two results may be combined into a 
simpler expression without any loss of indi- 
viduality. Since = cos cot + i sin ot where 
i = V ~ we have 
V(t) = [G(co2) + 2cuif(co2)] . if E{t) = 

A further simplification may be achieved by re- 
placing io by p, and G( — p^) + pH( — p^) by 
Y{p), so that 

V(t) = Y(p) • eP' if E(t) = eP' . (8) 

Y (p) is called the “steady-state transmission 
function’’ or just “transmission function’’ for 
short. 

Strictly speaking, (8) expresses the relation 
of steady-state response to signal only if p = io. 
However, it is customarily called a steady-state 
relation even when p is not a pure imaginary 
quantity. It may be noted that Y (p) is real 
when p is real. 

The simplicity of steady-state analysis de- 
rives from the fact that time occurs in the 
signal and throughout the network only in the 
form In particular, the determination of 
the transmission function is reduced to the 
solution of simultaneous algebraic equations 
which do not involve thq time factor. For a net- 
work in which the signal and the response are 
related by the linear differential equation (1) 
with constant coefficients, we obtain simply 

Y( \ _ CTo + aiP + • • • + OmV”' 

6o + 6ip -1- . . . + Kp- ' 

It may be noted that the poles of the transmis- 
sion function, also referred to as “infinite-gain 
points” in the p-plane, correspond to the roots 
of the characteristic function of the differential 
equation. Physical restrictions on the location 
of infinite-gain points will be considered in Sec- 
tion A.9. 


RELATIONSHIP BETWEEN 
IMPULSIVE ADMITTANCE AND 
TRANSMISSION FUNCTION 

A relationship between the impulsive admit- 
tance and the transmission function of a net- 


work may be obtained from (7). Putting 
E{t) = eP* when ^ > 0, we get 

V(t) = e-P" dr 

X oo 

W{t) e~P^ dr 

— W{t) dr . (9) 

The second term in (9) is a transient term due 
to the fact that we have taken E (t) =0 when 
^ < 0. The first term in (9), which involves the 
time only through is the steady-state term. 
Comparing this term with (8) we get 

F(p) = JJwit) e-P' dt (10) 

or, in the notation which will be introduced in 
the next section 

Yip) = L[Wit)] . (11) 

LAPLACE AND INVERSE LAPLACE 
TRANSFORMS 

The frequent use which is made of the 
Laplace transform and its inverse, in the 
analysis and design of fixed linear networks, 
warrants a brief discussion of these trans- 
forms. 

Given a function f{t) which is identically 
zero when ^ < 0, its Laplace transform g (p) is 
defined by the formula 

9(p) = wm = e-p‘ dt . (12) 

This is usually written with 0 for the lower 
limit, but by having the point ^ = 0 inside the 
range of integration, instead of at the end, we 
secure the same advantages for (12) that we 
gained in the case of (2) by having the point 
X = t inside the range of integration. Since f(t) 
is identically zero when ^ < 0 we could write 
— oo for the lower limit in (12), but this would 
run the risk of confusion with the so-called 
“bilateral Laplace transform.” On the whole, 
it is worth while to have a constant reminder 
that functions f(t) which are not identically 
zero when ^ < 0 are ruled out. 

The integral in (12) is usually not con- 
vergent for all values of p. That is, in order to 
secure convergence of the integral, it may be 
necessary to assume R{p) > where R(p) is 
the real part of p, and a is a real number. The 






APPENDIX A 


149 


result of the integration is a representation of 
g(v) in the half-plane R{v) > ct- Since the 
representation is analytic throughout the half- 
plane, the principle of analytic continuation 
allows us to extend the definition of g{v) to 
the remainder of the ^9-plane. 

Given a function g{v) which is analytic 
throughout the half -plane R{p) > c where c is 
a real number, its inverse Laplace transform 
f{t) is given by the formula 


transform of such a pulse over the interval 
0 <t<T is 

1 — e-p^ 
pT 

Hence 


L [5o(()] = lim 1 - ^ 

r^o pT 

F ormally therefore 


m 


= — r 

j,+ 


’c+too 

f^"‘[<7(p)] = .yb / g{p) e’’‘ dp (13) 

/c+ioo 


provided fit) is identically zero when t < 0. 
If the result of the integration in (13) is not 
identically zero when t < 0, gip) is not a 
Laplace transform and the application of the 
inverse transformation to it is meaningless. 


Translation Theorem 

A useful theorem can be established at this 
point. This is the translation theorem. 

If 

Gip) =LVFim 

then 

L-^[G(p)e-^«] = Fit- a) 
provided that Fit — a) =0 when t < 0. Trans- 
lation is to the right or left according as a is 
positive or negative. 

If it happens that Fit)=0 when t < t^ 
where to > 0, then the restriction is that 
a > — to. That is, a limited amount of transla- 
tion to the left is permissible. In general, to = 0 
and the restriction is therefore that a > 0. This 
theorem follows readily from (12) or (13). 

In all of the applications of (13) which we 
have any occasion to make in the analysis and 
design of fixed linear networks, the function 
gip) may be resolved into a sum of terms of 
the form Gip)e-p° where a > 0 and Gip) is a 
rational algebraic function with real coeffi- 
cients. Making use of the translation theorem, 
the problem of evaluating L-^lgip)] reduces to 
that of evaluating L-^[Gip)'\. Now, Gip) may 
be resolved into a sum of terms of the form 
pm or 1/ip — a ) where m = 0, 1, 2 . • • . We 
shall consider these two cases separately. 

The case G ip) = p^ will be treated by means 
of (12) and some limiting processes. In Sec- 
tion A.l the unit impulse was regarded as the 
limit of a rectangular pulse of duration T and 
amplitude 1/T. By means of (12) the Laplace 


L-i [1] = 5o(0 . (14) 

Similarly, the Laplace transform of a pulse 
over the interval a <t < a + T where a > 0 is 


Hence 


1 - e-p'^ 
pT 


e-pa 


L [6„(<- a)] = lim = 

T^O prp ^ 


e-pa ^ 


Formally therefore 


L-i [e-p«] = doit -a) . 

The last result follows directly from (14) using 
the translation theorem. 

Next, let 

diit) = lim ^0 (0 — ^0 (f — T) 

T-^O fp 


This is the limiting case, as shown in Figure 5, 
of two impulses of strengths 1/T and —1/T 
separated by a time interval T. It may be called 




Figure 5. An impulse doublet. 


an impulse of second order. By (12) and the 
previous results 


L [6i(<)] 


lim 1 — e 0'^ 

T^o p n • 


Formally therefore 

L-i [p] = 8, it) . 


(15) 


Proceeding in this fashion we may define an 
impulse of (m + l)th order as 


8mit) = lim ^ 

T-^O 


i(0 


it - T) 


(16) 


lONFIDEMTAL 


150 


APPENDIX A 


and we may then show that 

L . 

Formally therefore 

L-l M = dm(t) . (17) 

This disposes of the case G(p) = where 

m = 0, 1, 2 . . . . 

The case Gip) = 1/ {p — a) will be treated 
by means of (13) and Jordan’s lemma. 

Jordan’s Lemma 

If all the singularities of G(p) can be en- 
closed by a circle of finite radius with center at 
the origin, and if G(p) -> 0 uniformly with 
respect to arg z as \z\ oo, then 

G{p)e^dp~^ = 0 

where r is a semicircle of radius p, with center 
at the origin, to the right of the imaginary axis 
if t is negative, to the left of the imaginary axis 
if t is positive. 

By the use of this lemma the contour of inte- 
gration in (13) may be closed and the integra- 
tion may then be performed by the method of 
residues. In the case 


then 


fit) = £y^it - 

= fiir) ■ flit — t) dr . 


The functions /i(0 and fzit) are subject to 
conditions which permit the inversion of the 
order of integration in the following proof. 
However, these conditions are seldom of any 
concern. We have 


m = L-M^i(p) ’L[hm 


dp . 


Inverting the order of integration and noting 
that 


■j ^c + t’oo 

— / dp ~ ^ a \ > t 

= f(t - ifX<< 

we obtain the result stated in the theorem. 


5 ALTERNATIVE EXPRESSION OF THE 
RESPONSE-TO-SIGNAL RELATIONSHIP 


= (p _«)•»+! m = 0, 1, 2 • ■ • 

we readily obtain 

( 18 ) 


An important special case of (18), correspond- 
ing to a = 0, is 

<■»> 

Another useful theorem which is readily 
established by means of (12) and (13) is 
B Orel’s theorem. 


Borel’s Theorem 

If 9 (P) > 9i (P ) , 9 2 {p) are the Laplace trans- 
forms of f{t)y fiWy fzit), respectively, and if 

gip) = 9iip) g^ip) 


The result (8) obtained in Section A.2 sug- 
gests an operational expression of the form 

V{t) = Y{p) . E{t) (20) 

for the response-to-signal relationship what- 
ever the signal E{t) might be. If the equiva- 
lence of this operational expression to (2) is 
taken as a matter of definition we may readily 
discover the nature of the implied operation. 

In the light of Borel’s theorem, (2) may be 
expressed in the form 

L[V{t)] = L[Wit)] . Lm)] 

under the permissible assumption that 
when t < 0. Hence 

V{t) = L-i {L[Wit)] • L[Em 
or, by (11) 

V{t) = L-i [Y(p) •Lm)]] . (21) 

This is, therefore, in general the meaning of 
the operational expression (20).** 


We note that if S{p) = the operational 

expression 

V(t) = S{p) . Wit) 

is equivalent to (20). This form is used in Section 10.4 
and in Appendix B. 


fcOWFIDENTIA^ 


APPENDIX A 


151 


In very special cases of Y (p) it is possible 
and useful to give a more direct meaning to the 
operation (20). Consider the case Y (p) = p. 
We have, from the preceding section 


W(t) = L-i [p] = 


= lim 

T— >0 


8o{t) - doit - T) 
T 


Substituting into (7) we get 


V{t) = 


lim 

T-^O 


Eit) - E{t 
T 


T) 


dE(t) 

dt 


Hence, p stands for the derivative operator 
d/dt. Higher powers of p correspond to deriva- 
tives of higher orders because of (17) and 
(16). 

For the case Y (p) = 1/p we have, from the 
preceding section 


Wit) = [p-^ = 0 a t <0 

= 1 if t > 0 

substituting into (2) we get 

vit) r Ei\) dx . 

oo 


Hence, negative integral powers of p corre- 
spond to multiple integrations from —oo to t. 

These results are applicable to relations be- 
tween network characteristics. Suppose, for 
example, that 

Yip) = p^yip) 

and that 

Wit) = L-i [Yip)] and wit) = [yip)]. 

Then 


W(t) = -^w(t) 

and conversely w(t) is the m-fold integral of 
W (t) from —00 to 

Another special case of Y (p) to which a 
more direct operational meaning may be given 
is where a > 0. In view of the translation 
theorem we get 


Vit) = Eit - a) 


Perfect Prediction Operator 

The result obtained in the last paragraph 
suggests as a perfect prediction operator, 
where tf is the time of flight, since it gives the 
result that 

V it) = Eit tf) if t — tf 

Since it implies that a signal will be received 
before it is sent, however, it cannot be physi- 
cally realizable. It is nevertheless useful as an 
analytical tool in cases in which the signal has 
continuous derivatives of all orders in the 
closed interval ^ to ^ + tf, such as in the steady- 
state case considered in Section 13.1.4. 


^ TANDEM NETWORKS 

The transmission function Y (p) of a tandem 
combination of two networks whose individual 
transmission functions are Y^ip) and Y 2 (p), 
respectively, is given by 

Yip) = Y,ip) . Y,ip). (22) 

By Borel’s theorem, therefore, the impulsive 
admittance of the combination is given in terms 
of the individual impulsive admittances by 

=J^ Wi(t - X) • Wi{\) d\ . (23) 

The advantages of methods of network 
analysis and synthesis based upon steady-state 
properties are due partly to the essential sim- 
plicity of (22) compared with (23). These ad- 
vantages are particularly important in network 
synthesis or design. In this case Y (p) is a 
prescribed rational function of p. It is a simple 
matter to resolve it into two or more factors. 
A little experience in network design can go a 
long way toward a choice of factors which is 
the most favorable, on the whole, from the 
point of view of network conflgurations (in- 
cluding feedback networks) and element values. 
Redundant factors are easily introduced if they 
are desirable, as is done in the practical designs 
described in Section 13.1.5 and Section 13.3. 


"N 

SYMMETRICAL IMPULSIVE 
ADMITTANCES 


That is, the response is a retarded facsimile 
of the signal. This result is to be expected on 
the ground that corresponds to the trans- 
mission function of a properly terminated dis- 
tortionless, nondissipative, uniform transmis- 
sion line with delay a. 


Symmetrical impulsive admittances occur 
very frequently in the theory of data-smooth- 
ing networks. (See Section B.2.) The trans- 
mission functions corresponding to them pos- 
sess a property which we shall bring out in 
this section. 


* 


[ONFIDENTIAL 



J52 


APPENDIX A 


The symmetry of the impulsive admittance 
is expressed by 

W{T - t) = Wit) 

Since Wit) =0 when ^ < 0, it must be so also 
when t>T. Hence 

^T /2 ^T+ 

Yip) = / Wit)e-P^dt + / Wit)e-p^dt. 
c/o- */r /2 

By a change of variable of integration the sec- 
ond term may be expressed in the form 

<'7’/2 

WiT - t)e-p^'^-^^ dt 


£ 


or, because of the symmetry, in the form 

X Tl2 

Wit)eP^ dt . 

Hence, if the first term in Y (p) be denoted by 


we have 


X T 12 

Ti 


Wit)e~'^^ dt 


Yip) = Y,ip) -f Fi i-p) 

= [Fi(p)eP^/2 y^(_p)g-pr/2] g-pr/2 , 

At real frequencies {p = i(o) the bracketed fac- 
tor is evidently an even real function of w. 
Hence ' 


FM = Q(co2) . (24) 

Apart from discontinuities in the phase angle 
of the transmission function at real frequencies 
w for which is zero, the phase angle is 

proportional to frequency. Such a transmission 
function is referred to as a linear phase trans- 
mission function. Sinusoidal components of the 
signal, of frequencies less than the lowest fre- 
quency at which Q{(o^) vanishes, suffer phase 
retardations in transmission in proportion to 
their frequencies. These components therefore 
contribute no delay distortion. They are delayed 
by a uniform amount, just as they are in a 
properly terminated distortionless, uniform 
transmission line, although in the case of (24) 
they contribute amplitude or loss distortion 
through Q{o)^). The delay in (24) is just half 
of the '‘smoothing time'' T. 

SERIES RELATIONSHIPS BETWEEN 
IMPULSIVE ADMITTANCE AND 
TRANSMISSION FUNCTION 

Two useful series relationships between im- 
pulsive admittances and transmission functions 
will be derived in this section. 


Assume that W (t) admits the series expan- 
sion 

W(t) = A, + A^t + ... ... . (25) 

for small positive values of t. Then by (11) 
and (19) 

=~ + ji + --+^i + - • ( 26 ) 

If Ao 0 the transmission cannot drop off 
faster than 6 db per octave as the frequency 
increases indefinitely. If the transmission is to 
drop off ultimately at the rate of 6k db per 
octave all of the A's up to and including A^.o 
must be zero. This is to say that the impulsive 
admittance and all of its derivatives of orders 
up to and including the (k — 2)th must vanish 
at t = 0. 

Next, let us suppose that the impulsive ad- 
mittance and all of its derivatives of orders up 
to and including the (k — 2)th are continuous 
through all values of t including t = 0 except 
that the (k — 2)th derivative is discontinuous 
only at t = a. We may resolve the impulsive 
admittance into the sum Wj^(t) + Wo(t) where 
W^it) and all of its derivatives of orders up to 
and including the (k — 2)th are continuous 
through all values of t including t = 0, while 
W^it) =0 for all values of t < a. Then, for 
small positive values oft — a 

w,{l) = • (^‘-2^6) 

whence 

= (^ + 

Hence the transmission cannot drop off ulti- 
mately faster than ^(A: — 1) db per octave. We 
may summarize these results in the asymptotic 
loss theorem. 

Asymptotic Loss Theorem. 

If the transmission is to drop off ultimately 
at the rate of 6A; db per octave as the frequency 
increases indefinitely, the impulsive admittance 
and all of its derivatives of orders up to and 
including the {k — 2)th must be continuous 
through all values of t including t = 0. 

Discontinuities in Wit) or in some deriva- 
tive of Wit) cannot occur except at ^ = 0 in 
the case of physical lumped element networks. 
Practically, however, rapid changes in Wit) 


APPENDIX A 


153 


or in some derivative of W (t), at any value of 
t, may be expected to be associated with much 
the same behavior of the transmission at rea- 
sonably high frequencies. As an example con- 
sider the case 

Wii) = -e-^^ (^ > a > 0) 

V( ^ — d — Q: 

(p + a) (p + /?) * 


W (t) is continuous through t = 0 as long as /? 
is finite but becomes discontinuous there in the 
limit as oo. The first derivative of W (t) 
is discontinuous through t = 0 even when ^ is 
finite. The ultimate slope of the transmission is 
12 db per octave, in accordance with the 
asymptotic loss theorem, but in the range 
a < w < /3 the transmission appears to have a 
slope of only 6 db per octave. 

The importance of the observations made in 
the preceding paragraph, in the design of a 
network, is that if we attempt to approximate 
SL W (t) which has a discontinuity in a deriva- 
tive of lower order at t = a than at ^ = 0, the 
fact that the physical approximation must have 
continuous derivatives of all orders and through 
all values of t except t = 0 is not very signifi- 
cant. The ultimate slope of the transmission 
may not be reached until the frequency is too 
high to be of any importance. 

Another useful relationship between impul- 
sive admittance and transmission function fol- 


lows from the assumption that 


X 


(t) dt 


is finite for m = 0, 1, 2 ... If we expand the 
exponential in 


X co 

W(t)e-P‘ dl 


into a power series in pt we get 


Y(p)=M.-M,p + ^-^ + ... (27) 

where 

Mm = I t^W(t)dt . (28) 

The quantity Mm is the mth moment of the im- 
pulsive admittance. 

When Mo — 1 we speak of the response of the 
network as a weighted average of the impressed 
signal, and speak of the impulsive admittance 
W (t) as the weighting function. 


A9 PHYSICAL RESTRICTIONS ON THE 
TRANSMISSION FUNCTION 
The transmission function Y (p) of a lumped 
element network is a rational algebraic func- 
tion of p. It is real for real values of p (A.2) . 
Hence, the coefficients must be real, and there- 
fore the roots and poles must either be real or 
occur in conjugate complex pairs. 

Such a function may be expanded into the 
sum of a polynomial and a rational function 
whose numerator is of lower degree than the 
denominator. The latter may therefore be prop- 
erly expanded into partial fractions. For a 
partial fraction of the form 

"7 7- where m = 1 , 2 ... 

(p — 

the contribution to the impulsive admittance 
W (t) is by (18) 



For a pair of partial fractions of the form 

A iB A — iB 

{p — a A- ip — a — 

the contribution to the impulsive admittance is 
-1 

\m — 1)! + R sin ^t) . 

Since the impulsive admittance is the re- 
sponse to an impulsive signal it is clear that for 
a stable network the impulsive admittance must 
be free of terms which increase indefinitely 
with time, either on account of an amplitude 
factor of the form e"* where a > 0, or, in the 
event that oc = 0, on account of an amplitude fac- 
tor of the form where m > 1. Hence, the 
physical restrictions on the transmission func- 
tion are: 

1. No poles with positive real parts. 

2. Poles on the imaginary p axis must be 
simple.® 

The poles of a passive transmission function 
correspond to modes of free motion.^^^ Each of 
them may be shown^'^' to satisfy an equation of 
the form 

pT + F + - =0 
P 

where T, F, V are positive quantities whose 
values depend upon the particular mode and 


® Poles on the imaginary p axis must also be ruled 
out on the ground that persistent transients cannot be 
tolerated any more than growing transients. 


i 


ONFIDEM 




154 


APPENDIX A 


its activity. However, T is zero in the absence 
of kinetic energy, F is zero in the absence of 
energy dissipation, and V is zero in the absence 
of potential energy. It follows that in the 
absence of coils or in the absence of condensers, 
the transmission function must have poles only 
on the negative real p axis. 

For extremely narrow-band, low-pass appli- 
cations, such as data smoothing, it is not prac- 
ticable to build networks which call for coils 
because these generally turn out to be of many 
thousands of henries in inductance. The exclu- 
sion of coils from these applications does not, 
however, rule out transmission functions with 
complex poles. These may be realized with RC 
networks in feedback amplifier circuits as is 
shown in Chapter 12. 


QUASI-DISTORTIONLESS 
TRANSMISSION NETWORKS 


A quasi-distortionless transmission network 
is one which is distortionless only in a certain 
sense. This sense will be made clear in this 
section. 

Let 


Y('n) = 1 + QiP + + • • > 

^ 1 -f- 6ip -f b2P^ + • • • + bnP"^ ' 

This may also be written in the form 


(29) 


Y{p) = 1 -f cip + ^ H + 6f(p).(30) 

Obviously g(p) will be a rational function with 
the same denominator as Y {p) and a numera- 
tor of (n— l)th degree. If we now apply a sig- 
nal of the form 


E{t) =0 for < < 0 
= r for ^ > 0 

the response, by (21), will be 

V{t) = t’ + +•••+«■• 

+ r!L- ■[»(?)] (i>0)- 

If the coefficients in the rational expression for 
Y (p) are such that 


Cl = if, C2 = t}, ■ ■ Cr = t) (31) 

then 

V{t) = (<-}- tfY 4- r! L-i [g{p)] (<> 0). (32) 

The second term vanishes exponentially with 
time. The first term is an advanced or a re- 
tarded facsimile of the applied signal accord- 


ing to whether tf is positive or negative. We 
shall say that Y (p) is the transmission func- 
tion of a network which is quasi-distortionless 
to the signal 

Obviously a transmission network which is 
quasi-distortionless to the signal V' must also be 
quasi-distortionless to every signal V where s 
is a positive integer less than r, including zero. 
Hence we may state the quasi-distortionless 
transmission theorem. 

Quasi-Distortionless Transmission 
Theorem 

If the signal 

E{t) = 0 for < < 0 

= polynomial of degree r at most in t for 
t > 0 

is applied to a “quasi-distortionless transmis- 
sion network of order r,” the response will be 
of the form 

V{t) = E{t -f tf) + O(e“0 for < > 0, 

where 0(e-0 stands for terms which vanish 
exponentially with time. 

If tf>(^ the transmission network is a pre- 
dictor for polynomials of degree r at most. 
However, it does not begin to predict properly 
until some time has elapsed after the start of 
the signal, or of a new analytic segment of the 
signal; that is, until the transients have sub- 
sided sufficiently. 

If = 0 the transmission network may be 
regarded as a delay-corrected smoother for 
polynomials of degree r at most. This is ob- 
tained simply by taking 

fli = bi, a2 = b2, ar — br (33) 

in (29). 


All VARIABLE LINEAR NETWORKS 

A variable linear transmission network is 
one in which the response V (t) is related to the 
impressed signal E(t) by the linear differential 
equation (1) with coefficients which are pre- 
scribed functions of t The solutions of such a 
differential equation also obey the superposi- 
tion principle. Thus it is possible in this case 
also to formulate the response of the network 
to any signal in terms of its response to a 
standard impulsive signal. 

The response of a variable network to an 
impulse or any form of signal depends, how- 


APPENDIX A 


155 


ever, on the time at which the signal is applied. 
For an impulsive signal applied at time A. the 
response at time t will be represented by 
This is still called the “impulsive ad- 
mittance.” In the theory of linear differential 
equations it is known as a Green’s function. 
Physically, it must be identically zero for 
t<\. 

The superposition theorem may now be writ- 
ten in the form 


than those of fixed linear networks. This is due 
largely to the fact that there does not yet exist 
a technique corresponding to the steady-state 
and operational methods used in connection 
with fixed networks. However, there is a class 
of variable networks whose analysis and design 
are greatly facilitated by the fact that they are 
related to fixed networks by a transformation 
of the time variable. 

Consider the linear differential equation 


V{t) 




£(X) • W"(/,X) (ix 


(34) 




+ bn- 


dz^~^ 


-f- • • • +6; 


(W 

dz 


+ V = E 


provided the network has been properly de- 
signed and set into operation at t = 0. If 



for all values of ^ > 0, the response may be 
interpreted as a weighted average of the sig- 
nal. We note that in order to interpret the 
response as a weighted average of the signal, 
it is now no longer necessary to take the lower 
limit in (34) as — oo, as it was in the case of 
(2) for a fixed network. In other words, a 
variable network can be designed and set into 
operation at any time so that components of 
the signal which arrive before that time are 
completely ignored. 

The analysis and design of variable linear 
networks are in general much more difficult 


with constant coefficients. With appropriate 
restrictions on the roots of the characteristic 
function 

-f- + ••• + bi\ 4* 1 

it represents the response-to-signal relation- 
ship in a fixed network, if z is proportional 
directly to time. However, if 2 : is a more gen- 
eral function of the time, it will correspond to 
a variable network. The kind of transformation 
which is desired here is one which transforms 
the range — 00 < 2 : < + 00 into the range 
0 < ^ < + 00 with a one-to-one correspondence. 
Thus, we may take = log e(t) where 0(t) is a 
positive monotonic increasing function of t in 
the range 0 < ^ < + oo,with^!5Q 6(t) = 0. Sev- 
eral examples of 0{t), including 0{t) = t, are 
considered in detail in Chapter 14. 


APPENDIX B 


THEORETICAL MODIFICATIONS OF SMOOTHING FUNCTIONS TO FIT 
NONUNIFORM NOISE SPECTRA 


B est smoothing or weighting functions have 
been determined in Chapters 10 and 11 
under the assumption of random noise with flat 
spectrum. It has not been worth while in prac- 
tice to base the choice of best weighting func- 
tions on any more elaborate considerations of 
actual noise spectra, for at least three reasons : 

1. The effectiveness of a smoothing network 
shape of the weighting function. 

2. Noise spectra are subject to variations, 
due to factors which it is not desirable in prac- 
tice to attempt to control. 

3. Elaborate smoothing functions require 
elaborate networks with close tolerances on ele- 
ment values. 

Nevertheless, the theory of smoothing pre- 
sented in this monograph would not be com- 
plete without showing how more general shapes 
of noise spectra can be considered. Two meth- 
ods are presented here, which are generaliza- 
tions of those presented in Sections 10.3 and 
10.4, respectively. 


B 1 PHILLIPS AND WEISS THEORY^ 

Let g(t) be the tracking error, and W (t) the 
impulsive admittance of a smoothing and pre- 
diction circuit with smoothing time T. Then 
the error in prediction due to tracking error 
only, is 

V({) = jT g{t - r) ■ W{t) dr. 

The impulsive admittance W {r) will depend 
also upon the time of flight which, for purposes 
of analysis, is assumed to be constant. The 
mean square error is then 

Win) ■ C(ti - t2) ■ WiMndrt 

where 

C(x) = i g{\) ■ g(x + x) d\ ■ (1) 



C(x) is the autocorrelation of the error time- 
function g (a) . 

For ^ ni\i order smoothing and prediction 
circuit is now minimized with respect to the 
impulsive admittance under the restrictions^ 

T 

T^W{T)dr = {-ff'r (m = 0, 1, 2 ... n). (2) 

Hence W (r) must satisfy the integral equa- 
tion 




C(t — t) • W{T)dT = /co “h kit “h • • • ~h knP 

{0 < t < T) 


where the are constants to be determined. 
Now, if 



C{t - r) 


Wmir)dT = P (0 < f < T) 
(m = 0 , 1 , 2 ••• n) 


(3) 


then 


W{r) = koWoM + kiWiir) + ••• + knWnir). (4) 

The procedure is then to determine C(x) from 
(1), the Wmir) from (3), the k^n from (2) and 
(4), and Anally W {t) from (4). It may be 
noted that, in general, every k^i will be a poly- 
nominal of ni\i degree in tf. Hence the Wm(r) 
appearing here are not the same as those de- 
fined in Chapter 11, although W (t) should be 
the same if the same Wo(t) is used in Chapter 
11 . 

A difficulty of the theory given above is in 
the solution of the integral equations (3) . This 
difficulty is avoided in the theory given in the 
next section. However, the integral equations 
are easily solved in case of flat random noise, 
when C (x) is simply an impulse of strength K 
say, at a; = 0. Then 

Wmir) =~ 0 <T < T. 

Since the strength is irrelevant, it may be taken 
equal to T so that Wq{t) will be normalized. 


“• These follow from the discussions in Sections A.8 
and A.IO, especially equations (27), (28), (30), and 
(31). 


156 


i:u\eii)i;m l A 


APPENDIX B 


157 


For a linear prediction circuit it is then found 
that 

W{t) = 2 (2 + - |( 1 + 5 ) 

Putting T = 1 this may be expressed as 

W{t) = Wo{t) + G,i- (r) 

in terms of the G,„(t) and W^^r) of Section 
11.3. 


B 2 SYMMETRY OF BEST SMOOTHING 
FUNCTIONS 

The theory of Phillips and Weiss offers the 
most direct proof that the best smoothing or 
weighting function must be symmetrical, re- 
gardless of the noise power spectrum. The 
situation is that of minimizing ( 1 ) under only 
one of the restrictions ( 2 ), viz., the normaliz- 
ing condition 

W{r)dT = 1 (5) 

The weighting function is therefore deter- 
mined, up to a constant scale factor, by the 
condition that 

T 

C {t - t) • W{r)dr = k, (6) 

where A: is a constant. Substituting T — t for t 
and T — T for r, we have 

T 

C(t - t) • W{T - T)dT = k. (7) 

Since C( — x) = C(x), and since W (t) is de- 
termined uniquely by ( 6 ) and (5), it follows 
from ( 6 ) and (7) that 

W(T - t) = Wir). (8) 





GENERALIZATION OF ELEMENTARY 
PULSE METHOD 

The noise power transmitted through a net- 
work may be expressed in the familiar form 

P= I N(a;2) . |y(2co)|2^co 

where N (oy^) is the noise power spectrum and 
Y (p) is the transmission function of the net- 
work. Assuming that iV(w“) is a rational func- 
tion of 0 ) 2 , which is finite at all finite values of 
0 ) including zero, it is possible to determine a 


rational function S{p), which has no poles on 
or to the right of the imaginary axis in the 
p-plane with the exception of the point at infin- 
ity, and such that 

|,S(ia;)|2 = iV(co2). 

It may be readily shown that 

X co 

[F{t)Vdt (9) 

where F(t) is related to the impulsive admit- 
tance W {t) by the operational equation 

F{t) = S{p) • Wit) (10) 

The problem is now to minimize (9) under the 
restriction 

Wit)dt = 1 when /o > L (11) 



Let 


where 


Sip) = 


k 


Qip) 

Rip) 


Qip) = ip ai) (p + 0:2) • • • ip + Oim) 

Rip) = (p + ^1) ip W M • • • (p + M 

and k is of no consequence. One or more of the 
a’s, but none of the /3’s may be zero. Since the 
existence of the integral in (9) imposes the 
requirement that F{t) have no discontinuities 
of higher type than finite jumps in the range 
0 — < ^ < 00, the continuity conditions on W(t) 
in (10) must depend upon the difference be- 
tween m and n in the expressions for Q (p) and 
Rip). 

If m > ^^, it is fairly obvious that W (t) must 
be differentiable, in the ordinary sense, exactly 
m — n times. In other words, W it) and all its 
derivatives up to and including the {m — n 
— l)th must be continuous, but the (m — 9 ^)th 
derivative may have finite jumps. If m < we 
must consider the introduction into Wit) of 
discontinuities of higher type than finite jumps. 
These discontinuities arise in the formal ex- 
tension of the concept of differentiation to 
functions containing finite jumps. 

If a function 4>it) has a finite jump of am- 
plitude Ao at i = a, the value of 4>' it) at that 
point will be indicated formally as Aq • 80 ~ ct) 

where 80 it — a) is a unit impulse at ^ = a. If 
(j>'ia + 0 ) — <^'(a — 0 ) = Aj, the value of <t>"it) 
Sit t = a will be indicated formally as Ao • 8 ^ 
it — a) + Ai • 80 (^ — a) where 81 it — a) is a 


olNFTmv^mi 


158 


APPENDIX B 


unit doublet at t = a. And so on, for higher de- 
rivatives of (f>(t) . 

The expression (9) is a minimum under the 
restriction (11) if Wit) satisfies the differ- 
ential equation 

Qip) 'Qi - P) Wit) = const. (12) 


Case III. in 0, m < n) 

The 2m + 1 constants of integration in the 
general solution of (12) are first increased to 
2n -f 1 by appending the 2 (n — m) singularities 

6oW, 5i(/), ... 5n-n.-l (f) 


when 0 < t < 1 and Y ip) the condition 


J_ 

2x2. 


Sip) • Si — p) • Yip)e^^dp = const. 


when 0 < t < 1. 


(13) 


The restriction (11) itself requires that 
Wit) = 0 when t > 1, and 


8oit - 1), 8iit - 1), ... 8n-m-l it - 1) 

and then reduced to 2n by (14) . The remainder 
are determined by (13) or (15). 

In formulating 

Yip) = LlWit)] 



Wit)dt = 1. 


(14) 


it may be noted that 

L[5n(f - a)] = p^e-^P (a > 0) . 


Case I. in = 0) 

The general solution of (12) contains 2m + 1 
constants of integration which are determined 
by (14) and the 2m continuity conditions that 
Wit) and all of its derivatives up to and in- 
cluding the (m — 1) th must vanish at t = 0 and 
t = 1. 


Case II. in^ 0, m > n) 

The general solution of (12) contains 2m + 1 
constants of integration which are reduced 
to 2n in number by (14) and the 2(m — n) 
continuity conditions that Wit) and all of its 
derivatives up to and including the im — n — 
l)th must vanish at t = 0 and at t = 1. The 
remaining 2n constants are determined by (13) . 

The left-hand member of (13) may be for- 
mulated by the method of residues. The ex- 
pression for Y ip) should first be separated 
into two parts so that 

Yip) = YLip) + Ynip)e-P 

where Y^ip) and YRip) are rational functions 
of Sip)- Si—p)-YiXp)e^^ in the left-hand 
in the left-hand half of the p-plane for the first 
part of Y{p), and in the right-hand half for the 
second part. Hence, if the sum of the residues 
of S(p) • S( — p) • YL(p)e^* in the left-hand 
half of the p-plane be donated by Xl' and if the 
sum of the residues of Sip) • Si—p) • Ynip) • 
ep(i-i) in the right-hand half of the p-plane be 
denoted by Xr' then the condition (13) re- 
duces to 


Example of Case I 

Let S ip) = p^. The differential equation (12) 
requires W (t) to be a polynomial of degree 2m. 
The conditions at t = 0 require it to have a 
factor and those at t = 1, a factor (1 — t)^. 
This leaves only (14) to be satisfied. Hence 

(0 < ( 5 1) 

in agreement with (8) of Section 10.3. 


Example of Case II 


Let S{p) = ^ . (12) 

Then, by 

Wit) = i4o + + ^ 26 “' iO < t < 1). 

Hence 

Y{p) = — “ + -4L- + 


p p a p — a 

_r4» + 4ip+ ^1, 

[_p p + a: p — aj 


= 




2 / 3 = 




20 


20 J 


2^2 2/3 ^ 2/3 

Condition (15) is satisfied if 

A2 = i ^0 Q 


Si — 2ii = const. 


(15) 


APPENDIX B 


159 


where 


Example of Case III 


Q = 


0L2 — 02 


sinh | + cosh 


Hence 


1 + Q cosh “ 5 ) 

Wit) = ^ iO<t<l). 

1 + 2Q a 

a 2i 

j) 

In the limit as a 0, Sip) = — 

p -t P 

and 

1 + ^ <(!-<) 

Wit) = (0 < < < 1) . 

1 + 4-:^^ 


6 2 + ^ 

In terms of expressions (12), Section 11.3. 

W,(t) + kw,it) 


Wit) = 


(0 < / < 1) 


Let Sip) = 1/1 + 0. Then, by (12) and the 
rule for appending singularities in Case III 

U it) = Aq -|- Ai8oit) -|- A 28 oit — 1) iO ^ t ^ 1). 
Hence 

Y(p) = 4±.±AlP _ Ao - A 2P 
^ P P 

Ao Ao — 0Ai _g. 


V _ _J_ 

^2 1 - 20 ^ 


s« = - 


^0 — 0A2 
W2 


eW-i) . 


Condition (15) is satisfied if 

Ai = A2 = 

Hence 


0 


I k 

where k = l/6[0^/i2 + /?)]. This is reminis- 
cent of Stibitz’s results mentioned in Section 
10.3. 


1 + 


Wit) 


go(0 + 5o(^ - 1) 
0 


1 + 


(0 < / < 1) 




lONFFDENTIAL 







BIBLIOGRAPHY 


Numbers such as Div. 7-112.2-M9 indicate that the document listed has been microfilmed and that its title 
appears in the microfilm index printed in a separate volume. For access to the index volume and to the 
microfilm, consult the Army or Navy agency listed on the reverse of the half-title page. 


1 . 


2 . 


3. 


4. 


5. 


6 . 


7. 


8 . 


9. 


10 . 


11 . 


PART I 


* Chapter 2 

A Long-Range, High-Angle Electrical Antiaircraft 
Director, C. A. Lovell, NDCrc-127, Report to the 
Services 80, Bell Telephone Laboratories, Inc., June 
24, 1944. Div. 7-112.2-M9 

The Electric Antiaircraft Director T-10, W. S. 
Bowen, Research Project 1214, Antiaircraft Artil- 
lery Board, Fort Monroe, Virginia, Mar. 20, 1942. 

Div. 7-112.2-Ml 

Antiaircraft Dwector, T-15, OSRD 3009, OEMsr- 
353, Report to the Services 62, Western Electric 
Company, Inc., August 1943. Div. 7-112.2-M5 

The Antiaircraft Director, T-15-E1, OSRD 6410, 
OEMsr-353, Report to the Services 98, Bell Tele- 
phone Laboratories, Inc., July 30, 1945. 

Div. 7-112.2-Mll 

Curved-Course Antiaircraft Director Using Second 
Derivative Prediction, M. J. Kelley, OSRD 6291, 
OEMsr-1263, Report to the Services 103, Bell 
Telephone Laboratories, Inc., Oct. 1, 1945. 

Div. 7-112.2-M14 

Experiments on Curved Flight Computers, M. J. 
Kelley, OSRD 6569, NDCrc-178, Report to the 
Services 111, Bell Telephone Laboratories, Inc., 
Aug. 20, 1945. Div. 7-112.3-M2 

[Manual Rate Matching Mechanism for the M-7B1 
Director] Smoothing Attachment for M-7 Director, 
OEMsr-791, Bell Telephone Laboratories, Inc., Feb. 
14, 1944. Div. 7-112.2-M7 

Description and Operating Instructions for 
Smoother, T-1, OEMsr-899, The Bristol Company, 
November 1943. Div. 7-313.2-M2 

Description and Operating Instructions for Tenney 
Plotting Board, OEMsr-899, The Bristol Company, 
September 1943. Div. 7-112.4-M3 


12. A Study of Antiaircraft Tracking, John V. Atana- 

soff, Harold V. Gaskill, and others, OEMsr-165, 
Iowa State College. Div. 7-220.34-M3 

13. The RCA Computron, Jan Rajchman, OSRD 1538, 
OEMsr-591, Report to the Services 57, Radio Cor- 
poration of America, March 1943. Div. 7-112.4-M2 

14. Study of Design of Mechanical Dwector for 90-mm 

M-1 Gun (drawings included), Edwin L. Rose, 
OEMsr-1137, Bryant Chucking Grinder Company, 
Dec. 21, 1945. Div. 7-112.2-M16 

15. A Proposed Form of Two-Station Range Finder, 
Edwin C. Fritts, OEMsr-56, Problem DD-2492HH, 
Eastman Kodak Company, Dec. 5, 1944. 

Div. 7-210.19-M4 

16. Controlled Reticles for Lead Computing Gun Sights, 

Charles A. Morrison and Loyd A. Jones, OEMsr-56, 
Problem DD-2492L, Eastman Kodak Company, May 
8, 1944. Div. 7-112.11-M5 

17. Electronic Fire Control Computers. Intermediate 

Range Slant Plane Director, Arthur W. Tyler, 
Henry Harrison, and Fordyce E. Tuttle, OEMsr-56, 
Problems DD-2492C and 2492C-1, Eastman Kodak 
Company, October 1943. Div. 7-112.2-M6 

18. [The] 13y2-foot Superimposed Range Finder, Jo- 

seph Mihalyi and F. M. Bishop, OEMsr-56, Prob- 
lem DD-2492HH, Eastman Kodak Company, Nov. 
10, 1944. Div. 7-210.19-M2 

19. Full-Field Coincidence Range Finder of 15-inch 

Base Provided with Continuously Adjustable Range 
Compensation, Joseph Mihalyi and F. M. Bishop, 
OEMsr-56, Problem DD-2492R, Eastman Kodak 
Company, May 21, 1942. Div. 7-210.17-M2 

20. Short-Base Range Finders, Joseph Mihalyi, OEMsr- 

56, Problem DD-2492Q, R, X, DD, and II, Report to 
the Services 108, Eastman Kodak Company, Nov. 
19, 1945. Div. 7-210.17-M6 


Report Containing Description and Sketches of a 
Geometric-Type Predictor, William R. Smythe and 
Ira S. Bowen, California Institute of Technology, 
July 14, 1941. Div. 7-112.3-Ml 


21. Complementary Color Spotters for Machine Gun 
Tracer Fire, Joseph Mihalyi, OEMsr-56, Problems 
DD-2492Y and DD-2492Y1, Eastman Kodak Com- 
pany, Oct. 19, 1945. Div. 7-111-Ml 


Elements of Antiaircraft Fire Control (Final Re- 
port), Clifford G. Anderson, Clifford E. Berry, Sam 
Legvold, and William M. Stone, NDCrc-143, Iowa 
State College. Div. 7-112-Ml 


22. The Mark lU Illuminated Sight, Raymond W. Wen- 
gel, OSRD 6281, OEMsr-56, Problems 2492-GGl, 
2492-GG2, and others. Report to the Services 104, 
Eastman Kodak Company, [1946]. Div. 7-111-M2 


OiNFtDEMT^ 


161 


162 


BIBLIOGRAPHY 


23. Stereo Aid for Maxson Tiu'ret, Joseph Mihalyi, 

OSRD 6360, OEMsr-56, Problem DD-2492TT, Re- 
port to the Services 107, Eastman Kodak Company, 
Nov. 26, 1945. Div. 7-210.15-M5 

24. The T-28 Intermediate Range Director, Henry Har- 

rison, OSRD 6405, OEMsr-56, Service Project OD- 
142, Report to the Services 109, Eastman Kodak 
Company, Dec. 5, 1945. Div. 7-112.2-M15 

25. A Coil Yielding a Single Dipole Moment, Leon Bril- 
louin, OSRD 4020, OEMsr-1007, Study 120.1R, Re- 
port 188, AMG-Columbia, July 1944. 

Div. 7-112.4-M4 

26. On Sphei'ical Coils, Leon Brillouin, OSRD 4351, 

Study 120. IM, Report 288, AMG-Columbia, October 
1944. Div. 7-112.4-M5 

27. The M-SAlEl Modification of the M-5 or M-5A1 
Director for Intermediate Calibre Antiaircraft 
Guns, OSRD 1764, OEMsr-268, Report to the Ser- 
vices 60, Barber-Colman Company, June 1943. 

Div. 7-112.2-M4 

28. Investigation and Improvement of Intermediate 
Range Directors (Appendices Q through T), R. E. 
Schuette, D. L. Hall, and others, OEMsr-268, NDRC 
Project 31, Barber-Coleman Company, October 1945. 

Div. 7-112.2-M13 

29. Study ayid Experimental Investigation in Connec- 

tion with Computing and Servo Units for Range 
Finders, OSRD 617, General Motors Corporation, 
Inc., Apr. 1, 1942. Div. 7-321.1-M2 

30. An Antiaircraft Computing Sight, H. K. Weiss, 
OSRD 1765, OEMsr-883, Service Project OD-104, 
Report to the Services 61, Pitney-Bowes Postage 
Meter Company, August 1943. Div. 7-112. 11-M4 

31. Lead Computing Sights Based on an Angular Mo- 

mentum Invariant, Saunders MacLane, [OEMsr- 
1007], Study 55, Report 77, AMG-Columbia, Dec. 
10, 1943. AMP-703.1-M2 

32. Invay'iant Gyroscopic Lead Computing Sight, 

Arthur 1. Chalfant, OSRD 6406, OEMsr-1190, Re- 
port to the Services 110, Baker Manufacturing- 
Company, December 1945. Div. 7-112. 11-M8 

33. Description and Operating Instructions for Anti- 

aircraft Director, T-18, OEMsr-517, Bristol Com- 
pany, January 1943. Div. 7-112.2-M3 

34. Report on Computing Sight, T-62, OEMsr-892, Bar- 
ber-Colman Company, December 1944. 

35. Range Finder Problems, OEMsr-302, Polaroid Cor- 
poration, Nov. 30, 1944. Div. 7-210.19-M3 

36. Description of Ty'acking Head for Director, T-15, 
OEMsr-735, Barber-Colman Company, May 1944. 

Div. 7-112.2-M8 


37. A Dynamic Tester for Antiaircraft Directors, 

OEMsr-98, NDRC Project 25, Barber-Colman Com- 
pany, September 1945. Div. 7-112. 2-M12 

38. Instruction Manual [for the] Tape Dynamic Tester, 

Model 1, for Testing Antiaircraft Directors and 
Computers, OEMsr-904, Bell Telephone Laborato- 
ries, Inc., Nov. 30, 1944. Div. 7-312.2-M2 

39. histruction Book [for the] Data Recording System 

(Issue 3, Book 4), OEMsr-965, Bell Telephone Lab- 
oratories, Inc., Oct. 21, 1944. Div. 7-312.2-Ml 

40. Stibitz Computing Machine, Model B, R. R. Mon- 

roe, OEMsr-767, Department of Physics, University 
of North Carolina. Div. 7-312.3-Ml 

41. Mathematical Studies in Connection with the Design 

of Computers for Antiaircraft Fire Control, H. W. 
Bode and R. B. Blackman, NDCrc-178, Research 
Project 11, Bell Telephone Laboratories, Inc., Dec. 
15, 1945. Div. 7-112.3-M3 

Chapter 3 

1. Research Program on Servomechanisms, (manu- 

script only, bibliography appended), OSRD 38, 
Report to the Services 1, The Massachusetts Insti- 
tute of Technology. Div. 7-321.1-M6 

2. Behavior and Design of Servomechanisms, (bibliog- 
raphy appended), Gordon S. Brown, OSRD 39, 
Report to the Services 2, The Massachusetts Insti- 
tute of Technology, November 1940. 

Div. 7-321.1-Ml 

3. The Analysis and Design of Servomechanisms, 
(bibliography appended), Herbert Harris, Jr., 
OSRD 454, Report to the Services 23, The Massa- 
chusetts Institute of Technology. Div. 7-321. 1-M7 

4. The Analysis and Synthesis of Linear Servo- 
mechanisms, (bibliography appended), Albert C. 
Hall, OSRD 2097, Report to the Services 64, The 
Massachusetts Institute of Technology, May 1943. 

Div. 7-321. 1-M3 

5. Fundamental Theory of Servomechanisms, (bibliog- 
raphy appended), LeRoy A. MacColl, D. Van No- 
strand Company, 1945. 

6. ‘‘The Control of an Elastic Fluid,” (extensive biblio- 
graphical references), H. Bateman, reprinted from 
The American Mathematical Society Bulletin, 
September 1945. 

7. Resume of Research During the Year 19^.1 on 
Servomechanisms Used in Fire Control Equipment, 
OSRD 52, Report to the. Services 15, Oct. 7, 1941. 

Div. 7-101-Ml 

8. A Relay Controller to Provide Proper Fuze Time 

071 the Fuze Setter, M-8, Coyresponding to Director 
Fuze Range, John W. Anderson and Donald P. 
Campbell, The Massachusetts Institute of Tech- 
nology, July 1941. 7-112.4-Ml 


BIBLIOGRAPHY 


163 


9. Development of Hydraulic Booster Systems for 
Small Gu7is, Clifford Roberts, OSRD 51, OEMsr-18, 
Report to the Services 14, United Shoe Machinery 
Corporation, Sept. 16, 1941. Div. 7-111. 1-Ml 

10. Designs of Hydraulic Controls for Squall Caliber 

Guns, W. M. Sanderson, OSRD 510, OEMsr-173, Re- 
search Project 15, United Shoe Machinery Corpora- 
tion, Mar. 31, 1942. Div. 7-111. 1-M3 

11. Studies and Experimental hivestigatio7is in Con- 

nection with Hydr'aulic Mechanisms for Fire Con- 
trol, P. E. Nokes, George T. Hart, and others, 
OSRD 446, OEMsr-19, United Shoe Machinery Cor- 
poration, Feb. 28, 1942. Div. 7-111.1-M2 

12. Servomechanism Development, Milton Y. Warner, 
OEMsr-686, Westinghouse Electric and Manufac- 
turing Company, Inc., July 3, 1944. Div. 7-321. 2-Ml 

13. Control and Stabilization of Clutch Servos, Donald 

L. Hill, OEMsr-964, Barber-Colman Company, May 
1945. Div. 7-321.21-M2 

14. Hydraulic Remote Control for 37- and U0-7nm Gtm 
Mounts, OSRD 1763, OEMsr-522, Report to the 
Services 58, Research Project DIC-6047, The Mas- 
sachusetts Institute of Technology, June 1, 1943. 

Div. 7-321.2-M3 

15. Fmidamental Studies m Servomechanisms Rated 

Approximately 100 Watts, Volume 1, Hydra^dic 
Servos, OSRD 2098, Report to the Services 65, 
W-241-ORD-1142, Research Project DIC-6097, The 
Massachusetts Institute of Technology, September 
1943. Div. 7-321.2-M6 

16. Fundame7ital Studies m Se7'vomechanis7ns Rated 

Approximately 100 Watts, Volume 2, Elect7dc 
Se7'vos, OSRD 2099, Report to the Services 66, 
W-241-ORD-1142, Research Project DIC-6097, The 
Massachusetts Institute of Technology, September 
1943. Div. 7-321.2-M6 

17. Desc7'iptio7i and Operatmg histructions for Oil 

Geai's, M-SBl, When Used with the Re7note Control 
System, M-9 or M-10, OSRD 3000, Research Project 
DIC-6117, The Massachusetts Institute of Technol- 
ogy, Revised July 8, 1943. Div. 7-321.2-M4 

18. Experime7ital and Analytical Studies on Oil Gears, 
M-3B1, OSRD 3001, W-241-ORD-2592, Research 
Project DIC-6117, The Massachusetts Institute of 
Technology, September 1943. Div. 7-321. 2-M5 

19. [A] -’{OO-Cycle Frequency Controlled Motor-Gen- 

erator Set, OSRD 4693, OEMsr-1292, Report to the 
Services 86, Leeds and Northrup Company, Jan. 15, 
1945. Div. 7-321.21-Ml 

20. Data Transmission System Employing Voltage 

Dividers, Pilot Model, Bell Telephone Laboratories, 
Inc., Sept. 15, 1941. Div. 7-111.2-Ml 

21. Pe7'mutation Code Data Transmission System for 
Coast Artillery. Telegraph Systems, Bell Telephone 
Laboratories, Inc., November 1941. 

Div. 7-111.2-M3 


22. Seacoast Data Transmission Systein. Pilot Model, 

J. F. Quereau, OSRD 4294, OEMsr-404, Report to 
the Services 84, Leeds and Northrup Company, Oct. 
10, 1944. Div. 7-111.2-M2 

23. Gtm Ttirret Smoothness Tester, OEMsr-1185, 
Waugh Equipment Company, April 1946. 

Div. 7-323-Ml 

24. Network A7ialysis a7id Feedback Amplifier Design, 
Henry W. Bode, D. Van Nostrand Company, 1945. 

25. Bibliography [of the] Set'vo Panel Library on 

Servomechanis7ns, Section A, OSRD WA-5036-4, 

Enclosures 7 to 287/PR/253 Servo Panel Library, 
Ministry of Supply, [Great Britain], August 1945. 

Div. 7-321.1-M5 

26. Bibliog7'aphy [of the] Se7'vo Panel Library on 

Set'vomechatiistns, Section B, OSRD WA-5036-5, 

Enclosures 7 to 287/PR/253, Servo Panel Library, 
Ministry of Supply, [Great Britain], August 1945. 

Div. 7-321.1-M5 

Chapter 4 

1. Gyroscopic Lead Computing Sights, OSRD 50, Re- 
port to the Services 13, August 1941. 

Div. 7-112.11-Ml 

2. Accuracy of Lead Computation of Gyroscopic Lead 
Compfuting Sights When Used on Targets Flying a 
Straight Course, August 1942. Div. 7-112. 11-M3 

3. The Mark 23 Low Altitude Angular Rate Botnb- 

sight, OEMsr-504, NDRC Section 7.3, Research 
Project M-419, University of Michigan, Nov. 30, 
1945. Div. 7-122.1-M13 

4. [The] Gyroscopic Lead Computing Sight, Ma7'k 
15-P, C. F. Shriver, W. Bornemann, and Henry 
Harrison, OEMsr-56, Problem DD-2492KK, East- 
man Kodak Company, Oct. 30, 1945. 

Div. 7-112.11-M7 

5. Prelimmary l7istmictional Note on the Low Level 
Bombsight, Ma7'k III (Instructional Leaflet Inst. 
348), OSRD WA-629-1L, Royal Aircraft Estab- 
lishment, Great Britain, February 1943. 

Div. 7-122.1-M2 

6. Notes on Low Altitude Bombing: I Range Errors 
for Angular Depression and Angular Rate Meth- 
ods; II Effect of a Rangewise Imptact-Point Offset 
071 Range Errors for A7igular Depression and 
Angular Rate Methods; III Practical Evaluatioti of 
Composite Range Errors for Angidar Depression 
a 7 id Angidar Rate Methods; IV Range E7'rors for 
the Slant Range Methods; V Range Ei^rors for the 
Angular Rate Hybrid Method; VI Range Errors 
for Angular Depression and Angular Rate Meth- 
ods in Glide Bombing, Research Project 33, NDRC 
Section 7.2, Applied Mathematics Panel, and Frank- 
lin Institute, May 24 to Sept. 11, 1943. 

Div. 7-122.1-M3,4,5,6,7,8 


164 


BIBLIOGRAPHY 


7. The Evaluation of Integrals Arising in Exponential 
Delay Averages, OSRD 1839, Study 50. IR, Report 
43, AMG-Columbia, September 1943. 

Div. 7-313.1-M4 

8. Angular Rate Bombsight, Mark 23, C. F. Shriver, 

W. Bornemann, and Henry Harrison, OEMsr-56, 
Problem DD-2492QQ, Eastman Kodak Company, 
Oct. 25, 1945. Div. 7-122.1-Mll 

9. The Stabilized Angular Rate Bombsight, Mark 25, 

Model O, C. F. Shriver, W. Bornemann, and others, 
OEMsr-56, Problem DD-2492MM, Eastman Kodak 
Company, Oct. 29, 1945. Div. 7-122.1-M12 

10. Vacuum Regulator Valve, Irving T. Zuckerman, 

OEMsr-1366, Lawrence Aeronautical Corporation, 
Sept. 15, 1945. Div. 7-321.224-M3 

11. Torpedo Depth Control Development, OEMsr-1144, 
Foxboro Company, March 1946. Div. 7-321.222-M4 

12. Depth Engine, Irving T. Zuckerman, OEMsr-1366, 
Lawrence Aeronautical Corporation, Sept. 17, 1945. 

Div. 7-321.222-M2 

13. [A] 500-pound Torp>edo Regulating Valve, John 
F. Taplin, OEMsr-1366, Lawrence Aeronautical 
Corporation, Sept. 24, 1945. Div. 7-321.222-M3 

14. Camera Stabilizer, John F. Taplin, OEMsr-1366, 
Lawrence Aeronautical Corporation, Sept. 25, 1945. 

Div. 7-321.224-M5 

15. Puss Project, John F. Taplin and Bruce B. Young, 

OEMsr-1366, Lawrence Aeronautical Corporation, 
Sept. 25, 1945. Div. 7-321.223-Ml 

16. The Gyroscope Units for the PiloVs Universal 

Sighting System, C. F. Shriver, W. Bornemann, and 
others, OEMsr-56, Problem DD-2492VV, Eastman 
Kodak Company, Nov. 1, 1945. Div. 7-321.223-M2 

17. Pressure Reproducer, Irving T. Zuckerman, OEMsr- 

1366, Lawrence Aeronautical Corporation, Sept. 18, 
1945. Div. 7-321.224-M4 

18. Pneumatic Variable Resistor, Eugene Stolarik, 

OEMsr-1366, Lawrence Aeronautical Corporation, 
Aug. 20, 1945. Div. 7-321.224-M2 

19. A Reed-Controlled Speed Regulator for Air-driven 
Wheels, Henry Harrison, OEMsr-56, Eastman 
Kodak Company, Mar. 22, 1944. 

Div. 7-321.224-Ml 

20. Principles of Industrial Process Control, Donald P. 
Eckman, John Wiley & Sons, Inc., 1945. 

21. Industrial Instruments for Measurement and Con- 
trol, Thomas J. Rhodes, McGraw-Hill Book Com- 
pany, Inc., 1941. 

22. Pressure Drop in Tubing in Aircraft Instrument 
Installations, W. A. Wildhack, Technical Note 593, 
National Advisory Committee for Aeronautics, 1937. 


23. Aircraft Rate of Climb hulicators, Daniel P. John- 
son, Technical Report 666, National Advisory Com- 
mittee for Aeronautics, 1939. 

24. “Altitude Effects on an Uncompensated Rate of 
Climb Meter,” G. V. Schliestett, Journal of the 
Aeronautical Science, Vol. VI, No. 8, June 1939, pp. 
323-328. 

25. “Lag Determination of Altimeter Systems,” Richard 
M. Head, Journal of Applied Science, Vol. 12, No. 
1, January 1945. 

26. Inertia of Dynamic Pressure Arrays, Hans Wiede- 
mann, Technical Memorandum, 998, National 
Advisory Committee for Aeronautics, December 
1941. 

27. “Graphical Analysis of Delay of Response in Air- 
speed Indicators,” Dejuhasz, Journal of Applied Sci- 
ence, Vol. 10, No. 3, March 1943. 

28. Relay Devices and Their Application to the Solution 
of Mathematical Equations, Volume I Text, Vol- 
ume II Diagrams, H. Ziebolz, The Askania Regulator 
Company, 1940. 

29. “Equations Reduites pour le Calcul des Mouve- 
ments Amortis,” Pierre Curie, La Lumiere Elec- 
trique, t.XLI, Vol. 56, 1891, pp. 201, 270, and 307. 

30. Theory of Sound, Volume I, Lord Rayleigh, Lon- 
don, 1896. 

31. Dynamical Theory of Sound, H. Lamb, London, 
1925. 

32. Acoustics, G. W. Stewart and R. B. Linsay, D. Van 
Nostrand Company, 1930. 

33. Aj)plied Aco^istics, Harry F. Olsen and Frank 
Massa, The Blakiston Company, 1939. 

34. Theory of Vibrating Systems and Sound, I. B. 
Crandall, D. Van Nostrand Company, 1926. 

35. Introductory Pneumatic Analysis, Engene Stolarik, 

OEMsr-1366, Lawrence Aeronautical Corporation, 
Apr. 10, 1945. Div. 7-321.22-Ml 

36. Acoustic Design Charts, 1942, Frank Massa, The 
Blakiston Company. 

37. Pneuinatic Comjmting Devices for Lawrence Aero- 
nautical Corporatio7i, Ralph E. Byrne, Jr., Report 
I, Barber-Colman Company, May 9, 1945. 

Div. 7-321.22-M2 

38. Stability of Pneumatic Cup, Tube Systems, Ralph 

E. Byrne, Jr., Report II, Barber-Colman Company, 
May 28, 1945. Div. 7-321.22-M3 

39. Descriptioyi of the Fluid Gyroscojte, Calvin A. Gong- 

wer, Columbia University, New London Laboratory, 
Service Project NO-147, Division 6, Report 
D42/R651, Dec. 22, 1943. Div. 7-322.1-Ml 



BIBLIOGRAPHY 


165 


40. The Application of a Vibrating Rod as a Rate 
Measuring Device, OSRD WA-960-10, Report 
ARL/N-1/79.03-F, Admiralty Research Laboratory, 
Great Britain, Aug. 23, 1943. Div. 7-322. 2-Ml 

41. Investigation arid Improvement of Intermediate 
Range Directors (Appendices Q through T), R. E. 
Schuette, D. L. Hall, and others, OEMsr-268, NDRC 
Project 31, Barber-Colman Company, October 1945. 

Div. 7-112.2-M13 

42. Small Inclinometer, Irving T. Zuckerman, OEMsr- 

1366, Lawrence Aeronautical Corporation, April 1, 
1945. Div. 7-321.221-Ml 

Chapter 5 

1. Statistical Method of Prediction in Fire Control, 
Norbert Wiener, NDCrc-83, Research Project 6, 
Report to the Services 59, Dec. 1, 1942. 

Div. 7-112.2-M2 

2. The Extrapolation, Interpolation and Smoothing of 
Stationary Time Series with Engineering Applica- 
tions, Norbert Wiener, OSRD 370, Research Proj- 
ect DIC-6037, Report to the Services 19, The Mas- 
sachusetts Institute of Technology, Feb. 1, 1942. 

Div. 7-313.1-M2 

3. Notes on: The Extrapolation, Interpolation and 
Smoothing of Stationary Time Series, Norbert 
Wiener, Peter G. Bergmann, Research Project 
DIC-6037, Report to the Services 19, The Massa- 
chusetts Institute of Technology, Dec. 14, 1942. 

Div. 7-^13.1-M3 

4. The Extrapolation, Interpolation and Smoothing 

of Stationary Time Series with Engineering Ap- 
plications by Nobert Wiener, Digest of Manual by 
P. J. Daniell, OSRD Report 370, OSRD Liaison 
Office W-386-1, [1943]. Div. 7-313.1-M5 

5. An Exposition of Wiener's Theory of Prediction, 

Norman Levinson, OSRD 5328, OEMsr-1384, Note 
20, AMG-Harvard, June 1945. AMP-13-M21 

6. Relay Interpolator, S. B. Williams, OEMsr-1160, 
Bell Telephone Laboratories, Inc., Oct. 31, 1945. 

Div. 7-312.1-M4 

7. Relay Computers, George R. Stibitz, OSRD 4996, 

Report 171. IR, Applied Mathematics Panel, Feb- 
ruary 1945. Div. 7-312. 1-M2 

8. A Talk on Relay Computei's, George R. Stibitz, 

Memorandum 171. IM, Applied Mathematics Panel, 
March 1945. Div. 7-312.1-M3 

9. A Statement Concerning the Future Availability 

of a New Computing Device, George R. Stibitz, 
Note 7, Applied Mathematics Panel, November 
1943. Div. 7-312.1-Ml 

10. Ballistic Computer System (Instruction Book 
X-61877), Volumes 1 and 2 (final), June 1, 1944. 

11. Integrator Test Device and Polaroid-Type Torque 
Amplifier, J. G. Brainerd, OEMsr-856, University 
of Pennsylvania, Apr. 27, 1943. Div. 7-321.2-M2 


12. Some Experimental Results on the Deflection 

Mechayiism, Claude E. Shannon, Princeton Uni- 
versity, June 26, 1941. Div. 7-311-Ml 

13. Backlash m Overdamjjed Systems, Claude E. Shan- 
non. Div. 7-311-M3 

14. The Theory of Linear Differential and Smoothing 

Operators, Claude E. Shannon, Princeton Univer- 
sity, June 8, 1941. Div. 7-313.1-Ml 

15. A Height Data Smoothing Mechanism, Claude 

E. Shannon, May 26, 1941. Div. 7-313.2-Ml 

16. The Theory and Design of Linear Differential 

Equation Machines, Claude E. Shannon, Report 
to the Services 20, Bell Telephone Laboratories, 
Inc., January 1942. Div. 7-311-M2 

17. Prelimmary Report on the Study of Train Bomb- 
ing, OSRD 1869, OEMsr-65, Service Project AC-27, 
Applied Mathematics Panel Study ll.lR, Report 
to the Services 33, Princeton University, Aug. 25, 

1942. AMP-803.1-M1 

18. Preliminary Repoi't on Scatter Bombing, H. H. 

Germond, OSRD 904, Report to the Services 34, 
July 27, 1943. AMP-803.4-M1 

19. A Study of the Seriousness of the Effects, in the 

Planning and Executing of Bombing Missions, of 
Mis-Estimates of the Standard Errors of Aiming 
ayid Dispersion, OSRD 1149, Report to the Services 
46, Jan. 12, 1943. Div. 7-121. 1-Ml 

20. The Probabilities of Hitting, in Train Bombing, 

Rectangular Targets of Proportion One-by-Six or 
One-by-Nine, OSRD 1278, Report to the Services 
53, Mar. 10, 1943. Div. 7-121. 1-M2 

21. The Theory of Multiple Hits on Midtiple Targets 
in Train Bombing, OSRD 1476, Report to the Serv- 
ices 55 (Appendices A and B included). May 10, 

1943. Div. 7-121.2-Ml 

Chapter 6 

1. Instructions for Operating and Maintenance of the 
Experimental 10-cm Radio-Optical Range Finder, 
Mickey, NDCrc-156, Research Project 14, Bell 
Telephone Laboratories, Inc., [1942]. 

Div.7-210.13-M3 

2. Doppler Chronograph Development, Leon Katz, 
OSRD 4291, OEMsr-983, Service Project OD-100, 
Westinghouse Electric and Manufacturing Com- 
pany, Inc., East Pittsburgh, Pennsylvania, May 5, 

1944. Div. 7-324-Ml 

3. Handbook of Operating ayid Maintenance Instruc- 

tion for T-^ Chronograj)h covering the period from 
August 2, to December 31, 19A5, OEMsr-1405, 
Service Project OD-100, Report to the Services 101, 
Westinghouse Electric Corporation, Baltimore, 
Maryland, Oct. 10, 1945. Div. 7-324-M2 


166 


BIBLIOGRAPHY 


4. Prediction Mechanism for Torpedo Director for 

Destroyers and Light Cruisers, OEMsr-1208, Proj- 
ects NO-197 and NDRC-72, General Electric Com- 
pany, June 3, 1944. Div. 7-141-M2 

5. Torpedo Director, Mark 3^, for Motor Torpedo 

Boats, R. E. Coutant, OEMsr-1208, Service Projects 
NO-134 and NO-197, General Electric Company, 
Sept. 29, 1945. Div. 7-141-M3 

6. [Amplidyne Power Drives in One Unit (Redesign of 
Gun Director Mark 49)], OEMsr-1235, Servomecha- 
nisms Laboratory, The Massachusetts Institute of 
Technology. 

7. Letter to Professor Harold L. Hazen, Subject: Re- 
quest for Additional Funds for Gun Fire Control 
System Mark 56, I. A. Getting, July 27, 1945. 

Div. 7-112.2-MlO 

8. Development of Gun-Fire Control Mark 56, L. R. 
Lee, OEMsr-1299, Division 14, Report 497, Gen- 
eral Electric Company. 

9. Radiation Laboratory Report M-242; Preliminary 

Description of the Mark 56 Gun Fire Control Sys- 
tem, Walter R. Carmody and Albert D. Ehrenfried, 
OEMsr-262, Service Project NO-166, The Massa- 
chusetts Institute of Technology, Radiation Labora- 
tory, Dec. 15, 1945. . Div. 14-323.32-M7 


10. Final Report Contract OEMsr-lOAJf, for the period 
May 27, 19 U3 to October 31, 19^5, Service Project 
NO-166, Librascope, Inc. 

Part 1, History of the Contract and Patent Dis- 
closures, B. B. Willis, Oct. 31, 1945. 

Div. 7-122.3-Ml 

Part 2, Triangle Solver for Eagle Project Delta, 
D. C. Webster, Oct. 22, 1945. Div. 7-122.3-M2 

Part 3, Triangle Solvers for H2X Bombing Proj- 
ect Alpha, D. C. Webster, Oct. 18, 1945. 

Div. 7-122.3-M3 

Pay't U, Triangle Solver for Laboratory Use. 
Project Gamma, D. C. Webster, Oct. 18, 1945. 

Div. 7-122.3-M4 

Part 5, Redesign of Triangle Solver for Eagle 
Project Beta, D. C. Webster, Oct. 18, 1945. 

Div. 7-122.3-M5 

Part 6, Preliminary Ballistics Comjmter for a 
Gun Director System. Project Eta, Oct. 30, 1945. 

Div. 7-122.3-M6 

Part 7, Ballistic Computer, Mark U2, Model O 
Project Rho, D. C. Webster, Oct. 30, 1945. 

Div. 7-122.3-M7 

Part 8, Ballistic Comjmter, Mark U2, Model 1, 
Serial No. 1, D. C. Webster, Oct. 31, 1945. 

Div. 7-122.3-M8 


PART II 


1. The Extrapolation, Interpolation and Smoothing of 
Stationary Time Series with Engineering Applica- 
tions, Norbert Wiener, OSRD 370, Report to the 
Services 19, Research Project DIC-6037, The Mas- 
sachusetts Institute of Technology, Feb. 1, 1942. 

Div. 7-313.1-M2 

la. Ibid., Chapter 1. 

2. The Analysis and Design of Servomechanisms, 

Herbert Harris, Jr., OSRD 454, Progress Report to 
the Services 23, The Massachusetts Institute of 
Technology. Div. 7-321. 1-M7 

3. Behavior and Design of Servomechanisms, Gordon 
S. Brown, OSRD 39, Progress Report 2, The Mas- 
sachusetts Institute of Technology, November 1940. 

Div. 7-321.1-Ml 

4. Antiaircraft Director T-15, OEMsr-353, Report to 

the Services 62, Western Electric Company, Inc., 
August 1943. Div. 7-112.2-M5 

5. The Analysis and Synthesis of Linear Servomecha- 

nisms, Albert C. Hall, OSRD 2097, Report to the 
Services 64, The Massachusetts Institute of Tech- 
nology, May 1943. Div. 7-321. 1-M3 

6. Antiaircraft Director, T-15-E1, E. L. Norton, 
OEMsr-353, Report to the Services 98, Bell Tele- 
phone Laboratories, Inc., July 30, 1945. 

Div. 7-112.2-Mll 


7. Theoretical Calculation on Best Smoothing of Posi- 
tion Data for Gunnery Prediction, R. S. Phillips 
and P. R. Weiss, OEMsr-262, AMP Note 11, Re- 
port 532, The Massachusetts Institute of Tech- 
nology, Radiation Laboratory, Feb. 16, 1944. 

Div. 14-244.4-Ml 
AMP-703.4-M11 

8. A Long Range, High-Angle Electrical Antiaircraft 

Director [Final Report on T-10], C. A. Lovell, 
NDCrc-127, Research Project 2, Division 7 Report 
to the Services 80, Bell Telephone Laboratories, 
Inc., June 24, 1944. Div. 7-112. 2-M9 

9. Flight Records of Pitch, Roll, and Yaw, taken in 
a variety of bombers at Wright Field, Ohio, Sperry 
Gyroscope Company, 1942-5. 

10. Design and Performance of Data-Smoothing Net- 
work, R. B. Blackman, OEMsr-262, Report MM-44- 
110-38, [Bell Telephone Laboratories, Inc.], July 3, 
1944. 

11. Computer for Controllmg Bombers from the 
Ground, E. Lakatos and H. G. Och, OEMsr-262, 
July 24, 1944. 

12. A Position and Rate Smoothing Circuit for Ground- 
Controlled Bombing Computors, R. B. Blackman, 
OEMsr-262, Report MM-44-110-79, [Bell Telephone 
Laboratories, Inc.], Aug. 21, 1944. 


^NFIDENTL^ 


BIBLIOGRAPHY 


167 


13. A Two-Servo Circuit for Smoothing Present Posi- 
tion Coordinates and Rate in Antiaircraft Gun 
Directors, R. B. Blackman, Contract W-30-069- 
ORD-1448, Report MM-44-110-65, [Bell Telephone 
Laboratories, Inc.], Sept. 27, 1944. 

14. The Theory of Electrical Artificial Lines and Fil- 
ters, A. C. Bartlett, John Wiley and Sons, Inc., 
1931, p. 28. 

15. Network Analysis and Feedback Amplifier Design, 
H. W. Bode, D. Van Nostrand Company, 1945. 

15a. Ibid., Chapters 7, 8, 13, and 14 
15b. Ibid., p. 313. 

15c. Ibid., p. 326. 

15d. Ibid., p. 301. 

15e. Ibid., p. 33. 

15f. Ibid., p. 12. 

15g-. Ibid., p. 78. 

15h. Ibid., p. 110. 

15i. Ibid., p. 133. 

15j. Ibid., Chapter 5. 

16. Fundamental Theory of Servomechanisms, L. A. 
MacColl, D. Van Nostrand Company, 1945. 


17. Automatic Control Engineering, E. S. Smith, Mc- 
Graw-Hill Book Company, Inc., 1944. 

18. Die Lehre von den Kettenbriicken, B. G. Teubner, 
Leipzig, 1913. 

19. “Transient Oscillations in Wave Filters,’’ J. R. 
Carson and O. J. Zobel, Bell System Technical 
Journal, July 1923. 

20. “Harmonic Analysis of Irregular Motion,” Nor- 
bert Wiener, Joumial of Mathematics and Physics, 
Vol. 5, 1926, pp. 99-189. 

21. “Generalized Harmonic Analysis,” Norbert Wie- 
ner, Acta Mathematica, Stockholm, Vol. 55, 1930, 
pp. 117-258. 

22. “Stochastic Problems in Physics and Astronomy,” 
S. Chandrasekhar, Review of Modern Physics, Vol. 
15, 1943, pp. 1-89. 

23. “Mathematical Analysis of Random Noise,” S. O. 
Rice, Bell System Technical Journal, Vol. 23, 1944, 
pp. 282-332. 

23a. Ibid., Vol. 24, 1945, pp. 46-156. 


OSRD APPOINTEES 


DIVISION 7 

Chief 

Harold L. Hazen 

Technical Aides 

S. H. Caldwell J. R. Ragazzini 

Karl Wildes 

Consultants 

Gordon S. Brown Thornton C. Fry 


Members 


P. R. Bassett 
S. H. Caldwell 
Thornton C. Fry 
1. A. Getting 


Edward J. Poitras 
A. L. Ruiz 

Duncan J. Stewart 
Warren Weaver 


SECTION 7.1 

Chief 

Duncan J. Stewart 

Technical Aide 

G. R. Stibitz 


SECTION 7.2 

Chief 

S. H. Caldwell 

Technical Aides 

F. E. Martin George A. Philbrick 

Ruth Peters John B. Russell 

Hugh C. Wolfe 


Members 


C. G. Holschuh E. G. Pickels 

W. A. MacNair a. L. Ruiz 

G. E. Valley 


168 


rONFIDEMrAL 


OSRD APPOINTEES (Continued) 


SECTION 7.3 

Chief 

Edward J. Poitras 

Technical Aides 

Lawson M. McKenzie Edward J. Poitras 

Members 

George H. Pettibone John L. Taplin 

J. D. Tear 


SECTION 7.4 

Chiefs 

Preston R. Bassett Thornton C. Fry 

Technical Aide 

S. W. Fernberger 

Members 

Preston R. Bassett Theodore Dunham, Jr. S. W. Fernberger 

SECTION 7.5 

Chief 

Warren Weaver 

Technical Aide 

H. H. Germond 


George Agins 
R. E. Crooke 


SECTION 7.6 

Chief 

I. A. Getting 

Technical Aide 

Rogers Smith 

R. M. Page 
R. B. Roberts 


Members 
C. S. Draper 
A. W. Horton 
A. L. Ruiz 


)\FIDE>iTrAft 


169 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACT 


Contract 

Number 

Contractor 

Subject 

Division 

Project 

Number 

NDCrc-83 

Massachusetts Institute of Technology 
Cambridge, Massachusetts 

General mathematical theory of 
prediction and its applica- 
tions. 

7.5-6 

NDCrc-105 

Princeton University 

Princeton, New Jersey 

Mathematical studies relating 
to fire control. 

7.5-7 

NDCrc-116 

University of Wisconsin 

Madison, Wisconsin 

Mathematical studies. 

7.5-9 

NDCrc-127 
Supplement 4 

Western Electric Company 

New York, New York 

Electrical director (BTL-1). 

7.1-2 

NDCrc-156 

Western Electric Company 

New York, New York 

Optically tracked radio range 
finder. 

7.4-14 

NDCrc-163 

Massachusetts Institute of Technology 
Cambridge, Massachusetts 

Servomechanisms. 

7.3-1 

NDCrc-164 

California Institute of Technology 
Pasadena, California 

Geometrical predictor. 

7.1-4 

NDCrc-178 
Supplement 6 

Western Electric Company 

New York, New York 

Fundamental director studies. 

7.1-11 

NDCrc-186 
Supplement 3 

Princeton University 

Princeton, New Jersey 

Studies of fire control equip- 
ment. 

7.4-8 

NDRC-123 
Supplement 2 

California Institute of Technology 
Pasadena, California 

Methods of improving optical 
range finders. 

7.4-3 

OEMsr-19 

United Shoe Machinery Corporation 
Beverly, Massachusetts 

Hydraulic servomechanisms. 

7.3-16 

OEMsr-55 
Supplement 2 

Eastman Kodak Company 

Rochester, New York 

Height finder (Mihalyi). 

7.4-13 

OEMsr-56 
Supplement 6 

Eastman Kodak Company 

Rochester, New York 

Fire control research. 

7.1-17 

OEMsr-66 

Tufts College 

Medford, Massachusetts 

Psychological and physiological 
factors of importance in fire 
control. 

7.4-10 

OEMsr-98 
Supplement 5 

Barber-Colman Company 

Rockford, Illinois 

Dynamic tester (Barber-Col- 
man) . 

7.1-25 

OEMsr-165 
Supplement 1 

Iowa State College 

Ames, Iowa 

Analytical study of prediction 
devices and construction and 
test of such devices. 

7.1-12 

OEMsr-173 

United Shoe Machinery Corporation 
Beverly, Massachusetts 

Hydraulic controls for small 
caliber guns. 

7.3-15 

OEMsr-177 
Supplement 1 

Western Electric Company 

New York, New York 

Data transmission system. 

7.3-20 

OEMsr-184 
Supplement 1 

General Motors Laboratories, Inc. 

Chicago, Illinois 

Simplified electrical predictor. 

7.1-26 

OEMsr-253 

Massachusetts Institute of Technology 
Cambridge, Massachusetts 

Report on the extrapolation, in- 
terpolation and smoothing of 
stationary time series with 
engineering applications. 

7.5-29 

OEMsi'-268 
Supplement 6 

Barber-Colman Company 

Rockford, Illinois 

Intermediate range director 
M5A1E1. 

7.1-31 

OEMsr-302 
Supplement 4 

Polaroid Corporation 

Cambridge, Massachusetts 

Short base range finder. 

7.1-32 


]70 /©ONFIDENTIA 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACT (Continued) 


Contract 

Number 

Contractor 

Subject 

Division 

Project 

Number 

OEMsr-330 
Supplement 8 

Franklin Institute 

Bartol Research Foundation 

Philadelphia, Pennsylvania 

Airborne fire control equip- 
ment. 

7.2-33 

OEMsr-353 
Supplement 8 

Western Electric Company 

New York, New York 

OD-55, “one-plus” type BTL 
electric antiaircraft director. 

7.1-30 

OEMsr-404 
Supplement 2 

Leeds & Northrup Company 

Philadelphia, Pennsylvania 

Pilot model, data-transmission 
system. 

7.3-34 

OEMsr-444 
Supplement 1 

Franklin Institute 

Philadelphia, Pennsylvania 

Computations. 

7.5-39 

OEMsr-453 
Supplement 4 

Foxboro Company 

Foxboro, Massachusetts 

The effectiveness of controls 
and of data presentation. 

7.4-37 

OEMsr-473 
Supplement 1 

Dartmouth College 

Hanover, New Hampshire 

Effects of fatigue on space 
perception. 

7.4-36 

OEMsr-504 
Supplement 8 

University of Michigan 

McMath-Hulbert Observatory 

Ann Arbor, Michigan 

Gyroscopic rate applications. 

7.3-40 

OEMsr-517 
Supplement 3 

The Bristol Company 

Waterbury, Connecticut 

Rocket director development. 

7.1-38 

OEMsr-522 
Supplement 3 

Massachusetts Institute of Technology 
Cambridge, Massachusetts 

Improvement of servo for 37 
and 40 mm guns. 

7.3-35 

OEMsr-555 
Supplement 5 

Harvard University 

Cambridge, Massachusetts 

Acuities in telescopic vision. 

7.4-43 

OEMsr-562 
Supplement 5 

American Gas Association 

Testing Laboratories 

Cleveland, Ohio 

Helium retentivity. 

7.4-41 

OEMsr-581 

Tufts College 

Medford, Massachusetts 

Relation between fatigue and 
tracking. 

7.4-42 

OEMsr-591 

Radio Corporation of America 

Princeton, New Jersey 

Electronic computing devices 
for predictors. 

7.1-48 

OEMsr-618 
Supplement 2 

Columbia University 

New York, New York 

Air warfare analysis. 

7.5-47 

OEMsr-637 
Supplement 1 

Ohio State University 

Columbus, Ohio 

Stereoscopic acuity. 

7.4-45 

OEMsr-648 
Supplement 1 

Stanolind Oil & Gas Company 

Tulsa, Oklahoma 

Fire control analysis device. 

7.2-49 

OEMsr-686 
Supplement 3 

Westinghouse Electric and 

Manufacturing Company 

Pittsburgh, Pennsylvania 

Servos for medium -caliber 

guns. 

7.3-46 

OEMsr-732 
Supplement 6 

University of Texas 

Austin, Texas 

Testing plane-to-plane fire con- 
trol equipment. 

7.2-50 

OEMsr-735 
Supplement 1 

Barber-Colman Company 

Rockford, Illinois 

Combined tracking and range- 
finding devices. 

7.1-52 

OEMsr-767 
Supplement 4 

University of North Carolina 

Chapel Hill, North Carolina 

Antiaircraft fire control test- 
ing. 

7.1-54 

OEMsr-780 

Wilcolator Company 

Elizabeth, New Jersey 

Gyroscopic computers. 

7.3-55 

OEMsr-784 

Bausch & Lomb Optical Company 

Rochester, New York 

Invar bar for M2 height finder. 

7.4-56 


i[fi(^FIDENTiAS 


171 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACT (Contimied) 


Contract 

Number 

Contractor 

Subject 

Division 

Project 

Number 

OEMsr-791 
Supplement 1 

Western Electric Company 

New York, New York 

Modification of M7 director for 
field conversion. 

7.1-51 

OEMsr-817 
Supplement 1 

University of California 

Los Angeles, California 

Statistics of train bombing. 

7.5-23-2 

OEMsr-818 
Supplement 1 

Columbia University 

New York, New York 

Statistics of train bombing 

7.5-23-1 

OEMsr-856 
Supplement 1 

University of Pennsylvania 

Philadelphia, Pennsylvania 

Investigation of Bush differen- 
tial analyzer. 

7.5-62 

OEMsr-883 
Supplement 2 

Pitney-Bowes Postage Meter Company 
Stamford, Connecticut 

Development of computing sight 
T31. 

7.1-61 

OEMsr-892 
Supplement 2 

Barber-Colman Company 

Rockford, Illinois 

Field artillery antitank fire 
control equipment. 

7.3-59 

OEMsr-899 
Supplement 2 

The Bristol Company 

Waterbury, Connecticut 

Chart type data smoother and 
retransmitter. 

7.1-64 

OEMsr-904 
Supplement 8 

Western Electric Company 

New York, New York 

Punched tape dynamic tester. 

7.1-60 

OEMsr-952 
Supplement 2 

Eastman Kodak Company 

Rochester, New York 

Range finder redesign. 

7.4-58 

OEMsr-964 

Barber-Colman Company 

Rockford, Illinois 

Servomechanisms. 

7.3-27 

OEMsr-965 
Supplement 3 

Western Electric Company 

New York, New York 

Test data recorder. 

7.1-63 

OEMsr-983 
Supplement 6 

Westinghouse Electric and 

Manufacturing Company 

Pittsburgh, Pennsylvania 

Study of employment of radio 
doppler effect for velocity 
determination of projectiles. 

7.6-65 

OEMsr-991 
Supplement 7 

Jam Handy Organization, Inc. 

Detroit, Michigan 

Vector gunsight and assessing 
camera. 

7.2-67 

OEMsr-992 
Supplement 5 

General Electric Company 

Schenectady, New York 

Airborne gunnery computers. 

7.2-57 

OEMsr-1008 

Bausch & Lomb Optical Company 

Rochester, New York 

Tank fire control. 

7.4-66 

OEMsr-1016 
Supplement 1 

Bausch & Lomb Optical Company 

Rochester, New York 



OEMsr-1044 
Supplement 8 

Librascope, Inc. 

Burbank, California 

Development of computer for 
gun director Mark 56. 

7.6-85 

OEMsr-1059 
Supplement 3 

Brown University 

Providence, Rhode Island 

Reticle design. 

7.4-44 

OEMsr-1116 
Supplement 1 

Keuffel & Esser 

New York, New York 



OEMsr-1137 
Supplement 3 

Bryant Chucking Grinder Company 

Springfield, Vermont 

Mechanical director for 90 mm 

AA guns. 

7.1-68 

OEMsr-1144 
Supplement 3 

Foxboro Company 

Foxboro, Massachusetts 

Development of steering mech- 
anisms for torpedoes. 

7.3-69 

OEMsr-1160 

Western Electric Company 

New York, New York 

Relay interpolator. 

7.5-70 

OEMsr-1181 

General Electric Company 

Schenectady, New York 

Gyro unit for gun director 
Mark 56. 

7.6-71 

172 




CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACT (Continued) 


Contract 

Number 

Contractor 

Subject 

Division 

Project 

Number 

OEMsr-1185 
Supplement 3 

Waugh Equipment Company 

New York, New York 

Mechanism to measure the 
smoothness of control of air- 
craft turrets. 

7.3-75 

OEMsr-1190 
Supplement 6 

Baker Manufacturing Company 

Evansville, Wisconsin 

Course-invariant sights. 

7.1-73 

OEMsr-1208 
Supplement 3 

General Electric Company 

Schenectady, New York 

Sea-borne torpedo director. 

7.6-72 

OEMsr-1235 

Massachusetts Institute of Technology 
Servomechanisms Laboratory 
Cambridge, Massachusetts 

Redesign of gun director Mark 

49. 

7.6-77 

OEMsr-1236 

Western Electric Company 

New York, New York 

Automatic relay ballistic com- 
puters. 

7.5-74 

OEMsr-1237 
Supplement 3 

Columbia University 

New York, New York 

Fire-control electronics. 

7.2-76 

OEMsr-1263 
Supplement 5 

Western Electric Company 

New York, New York 

Development of computer T17. 

7.1-78 

OEMsr-1276 
Supplement 4 

Northwestern University 

Chicago, Illinois 

Aircraft fire control analysis. 

7.2-80 

OEMsr-1292 
Supplement 2 

Leeds & Northrup Company 

Philadelphia, Pennsylvania 

Speed regulator for motors and 
motor generators. 

7.3-81 

OEMsr-1299 

General Electric Company 

Schenectady, New York 

Development of gun director 
Mark 56. 

7.6-79 • 

OEMsr-1366 
Supplement 4 

Lawrance Aeronautical Corporation 

Linden, New Jersey 

Control elements for fire con- 
trol applications. 

7.3-82 

OEMsr-1387 
Supplement 2 

The Bristol Company 

Waterbury, Connecticut 

Components for pilot operated 
sight. 

7.2-84 

OEMsr-1405 

Westinghouse Electric Corporation 
Baltimore, Maryland 

Study of employment of radio 
doppler effect for velocity de- 
termination of projectiles. 

7.6-83 

Symbol Numbers 
1891, 

1938 

International Business 

Machine Corporation 

New York, New York 

Torpedo director. 

7.2-53 


173 


SERVICE PROJECT NUMBERS 



The projects listed below were transmitted to the Executive Secre- 
tary, NDRC, from the War or Navy Department through either 
the War Department Liaison Officer for NDRC or the Office of 

Research and Inventions (formerly the Coordinator of Research 
and Development) , Navy Department. 

Service 

Project 

Number 

Subject 

AN-4 

Angular velocity bombsight for use at low altitudes. 

AN-5 

Airborne fire control committee. 

• AC-27 

Bombardier’s calculator. 

AC-28 

Hydraulic controls for small caliber guns. 

AC-47 

Development of equipment for testing fire control apparatus. 

AC-61 

Development and construction of an own-speed computer. 

AC-116 

Determination of the most suitable method of testing flexible aerial gunsights. 

AC-119 

Procurement of one set of film assessment equipment. 

AC-121 

Airborne rocket* sights. 

AC-128 

Airborne computer calibration. 

OD-20 

Development of a short-base wide-angle range finder suitable for use in moving vehicles. 

OD-28 

Electrical computing director for the 90 mm AA gun. 

OD-41 

Transmission of data from base end stations. 

OD-51 

Self collimating height finder. 

OD-53 

Reduction in errors in remote control system Ml. 

OD-55 

“One-plus” type BTL electric antiaircraft director. 

OD-56 

Director for use with rocket projectors. 

OD-67 

Field artillery antitank fire control equipment. 

OD-78 

Investigation of Bush differential analyzer. 

OD-89 

Use of nonreflecting films on height finder optics. 

OD-91 

Modification of director M7. 

OD-93 

Computing sight T20. 

OD-100 

Study of employment of radio doppler effect for velocity determination of projectiles. 

OD-101 

Development of a new type stereoscopic range finder. 

OD-104 

Development of computing sight T31. 

OD-105 

Test data recorder. 

OD-111 

Triangle solver. 

OD-120 

Intermediate range director M5A1E1. 

OD-122 

Smoothing device for employment with SCR-268. 

OD-127 

Extend the utility of the Barber-Colman dynamic tester. 

OD-132 

Plotting board T9. 

OD-142 

Intermediate caliber antiaircraft range director T28. 


SERVICE PROJECT NUMBERS (Continued) 


Service 

Project 

Number 

OD-153 

OD-157 

OD-203 

D2-40 

NA-114 

NA-136 

NA-158 

NA-161 

NA-168 

NA-203 

NA-232 

NO-106 

NO-107 

NO-108 

NO-109 

NO-112 

NO-113 

NO-122 

NO-128 

NO-129 

NO-130 

NO-131 

NO-134 

NO-136 

NO-137 

NO-139 

NO-145 

NO-152 

NO-154 

NO-166 

NO-178 

NO-179 

NO-180 

NO-186 

NO-190 

NO-191 

NO-197 

NO-203 


Subject 


Automatic relay ballistic computers. 

Development of computer T17. 

Blind firing installation for 40 mm gun using director and radar. 

Gyroscopic director development. 

Construction of two machines for testing aircraft fire control equipment. 

Adaptation of an existing electronic remote control system to a Bell Sunstrand hydraulic 
turret drive unit. 

Mechanism to measure the smoothness of control of aircraft turrets. 

Aircraft fire control analyses — Naval Air Station, Patuxent River. 

Slant range computer. 

Design of computing mechanism for the torpedo deflection trainer. 

Development of Razon attachment for Navy 7-A-3 Trainer. 

Aircraft torpedo director. 

Electronic computer. 

Probability and statistical study of plane-to-plane fire. 

Methods for operating equipment related to fire control apparatus. 

Short-base antiaircraft range finders. 

Directing tracer fire in plane-to-plane firing. 

Development of a gyro lead-computing tail turret gunsight. 

Projected plane gunsight. 

Antisubmarine bombsight. 

(No title available.) 

Probability studies. 

Mark 32 (TBF type) torpedo director for motor torpedo boats. 

(No title available.) 

Comparative study, three types of reticles. 

Stereoscopic range finder. 

(No title available.) 

Development of gunsight computer for flexible aerial gunnery. 

(No title available.) 

Development of gun director Mk 56 (Division 14). 

Tracking tests on optical ring sights. 

Development of steering mechanisms for torpedoes. 

Maneuverable target for Mark 2 bombing trainer. 

To provide target course stabilization for the torpedo director Mk 30. 

Airship bombsight. 

A preset attachment for the Mark 15 bombsight. 

Design of fire control for torpedoes for destroyers and light cruisers. 

Redesign of gun director Mark 49. 





175 


SERVICE PROJECT NUMBERS (Continued) 


Service 

Project 

Number 

Subject 

NO-207 

Improvements to the gun director Mark 37. 

NO-209 

Stabilized roll indicator (with Section 6.1). 

NO-216 

Adaptations of gunsight Mk 18 for other purposes. 

NO-240 

NDRC advisory service on radar bombsight. 

NO-241 

Small combination aircraft torpedo director and low altitude bombsight. 

NO-242 

Range-type torpedo director with provision for servo range input. 

NO-243 

Torpedo bombsight for high altitude torpedo release. 

NO-244 

Mechanization of a correction for maneuvering targets. 

NO-255 

Illuminated sight Mk 14. 

NO-265 

Multiple purpose pilot’s sighting system. 

NO-268 

Study of aircraft turrets equipped with lead computing sights and aided tracking. 

NS-334 

Advisory services of Division 7 for Bureau of Ships. 


176 



INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. For 
access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


AAB (Antiaircraft Board) com- 
puter, 57-58 

Admittance of linear networks, im- 
pulsive 

see Impulsive admittance of lin- 
ear networks 

Air warfare analysis project, 60- 
61 

Aircraft, constant-velocity, 129-133 
ground-control bombing com- 
puter, 132-133 

position data smoothing, 131 
target paths, circular, 129-130 

Aircraft, physical limitations, 82- 
83 

Aircraft turret smoothness tester, 
43 

Analytic arcs assumption, target 
courses, 100-106 

calculation of best smoothing 
• time, 104-105 

nonlinear and variable systems, 
106 

Poisson distribution of segment 
end points, 101-102 
probability distribution of future 
positions, 102-104 
summary, 100-101 
weighting functions, 104 

Angular rate bombsight Mark III, 
46 

Antiaircraft Board (AAB) com- 
puter, 57-58 

Antiaircraft fire-control system, 
data smoothing 

analytic arcs assumption, target 
paths, 100-106 
autocorrelation, 77-78 
extrapolation, 75-76 
filters, 79-81, 89-93 
human factors, 83-84 
least squares assumption, 78-79, 
104 

mathematical formulae, 54-61, 
107-142, 156-159 
noise functions, 87-89 
physical limitations of aircraft, 
82-83 

predicting theory, 54-56, 75-76, 
85, 97-98, 127-130 
relation between data smoothing 
and prediction 73 
“settling time”, 83 


/ 

signal spectrum of target path, 
86 

summary, 71-73 

tactical criteria for evaluation, 73 
time series analysis, 76-77, 81 
variable and nonlinear circuits, 
134-143 

Wiener’s prediction theory, 93-97 
Antitank computing sight T62; 32 
Attenuation-phase relations for 
physical networks, 85, 92, 
97-98 

Autocorrelation method for data 
smoothing and prediction 
derivation, 78 

least squares assumption, 78-79 
settling time, finite, 109-110 
statistical significance, 78-79 
Automatic calculating devices, 56- 
59 

ballistic computer, 57-58 
differential analyzer, 58-59 
relay interpolator, 57 
tape dynamic testers, 56 
Automatic control mechanisms 
see Servomechanisms 

Baker course-invariant computing 
sight, 8, 28-29 

Ballistic computer, 35, 57-58 
Barber-Coleman Co. 
dynamic tester, 6, 7, 33 
gyroscopes, 52 

M5 director, modification, 25-28 
range finder, 33 
servos, clutch-type, 39 
Bell Telephone Laboratories 
ballistic computer, 35, 57-58 
dynamic tester, 6, 7 
gun director, 13-15 
radar, 62 

relay interpolator, 57 
sea coast data transmission, 42 
Bombardier’s calculator, 60 
Bombing computer, ground-control, 
119, 132-133 
Bombsights 

British, angular rate Mark III, 
46-47 

Navy, Mark 23; 9, 46-49 
Navy, Mark 25; 9, 49 
Boosters, hydraulic, 39 
BoreTs theorem, linear network 
theory, 150 


Bristol Company, smoothing de- 
vices for directors, 17-19 
British angular rate bombsight 
Mark III, 46 

Bryant Chucking Grinder Co., fire- 
control mechanism, 8, 21 

Calculating devices, automatic 
see Computers 

California Institute of Technology, 
predictor design, 19-20 
Camera stabilizers, aerial, 51 
Chronograph T4, radar, 63-65 
Columbia University Division of 
War Research, fluid gyro, 52 
Communication engineering, data 
smoothing, 79-81 
Computers 

ballistic, 35, 57-58 
bombing, ground-control, 119, 132- 
133 

curved flight, 15-16 
differential analyzer, 58-59 
electronic, 20-21 
mech'anical aids, 56-59 
relay interpolator, 57 
tape dynamic testers, 56 
Computing sights 
antitank, 32 

course-invariant, 8-9, 28-29 
gyroscopic lead-computing, 9, 44- 
46 

Mark 15; 45-46 
Mark 15-P, 46 

Constant-velocity aircraft, 129-133 
ground-control bombing com- 
puter, 132-133 
position data smoothing, 131 
target paths, circular, 129-130 
Course-invariant computing sights, 
8, 28-29 

Curved flight computers, 15 
Curve-fitting method, data smooth- 
ing, 108-109 

Data smoothing, antiaircraft fire- 
control 

see Antiaircraft fire-control sys- 
tem, data smoothing 
Differential analyzer, ballistics, 58- 
59 

Dynamic testers, error computa- 
tion, 33-35 


178 


INDEX 


Eastman Kodak Co. 

lead-computing sights, 9, 45 
range finder, 32 
T28 director, 8 

Elementary pulse method, data 
smoothing, 110, 157-159 

Exponential smoothing circuit, 
data smoothing, 107-108 

Extrapolation method, statistical 
prediction, 54, 56, 75-76, 112 

Feedback amplifiers, transmission 
functions, 119-121, 130-131, 
134-135 

Filters, data smoothing 
communications, 79-81 
lag, 91-93 
theory, 89-90 

Fire control, 3-37, 44-69, 81-84, 86 
administrative operations, 10-11 
antiaircraft weapons, 6-9, 19-25 
apparatus, 3-4, 6-8 
automatic weapons, 8, 25-30, 36- 
37 

chronograph T4 radar, 63-65 
gun directors, 13-19, 65-69 
gyroscopic lead-computing sights, 
44-46 

mathematical analysis of prob- 
lems, 54-61 

Navy bombsight Mark 23; 46-49 
plane-to-plane, 6-8, 61 
pneumatic techniques, 9, 44-53 
prewar status, 5-6 
recommendations for future re- 
search, 6-10 
SCR-547 radar, 62-63 
signal spectrum, 86 
summary, 3-7 
testing program, 33-36 
torpedo directors, radar, 65-66 
tracking, 81-84 

Fire-control system, data smooth- 
ing 

see Antiaircraft fire-control sys- 
tem, data smoothing 

Foxboro Co., torpedo control, 9, 
50-51 

Franklin Institute, fire-control de- 
vices, 9, 60 

General Electric Company, fire- 
control devices, 9, 66-69 

General Motors Laboratories, elec- 
trical lead-computing device, 
28 

Green’s function, impulsive admit- 
tance, 147, 155 

Gun directors 

M7 gun director (Weissight), 6, 
17, 28 


M9 antiaircraft, 6, 8, 13-14, 16, 
125-131 
Mark 49; 66 
Mark 56; 8, 66-69 
range, intermediate, 24-25 
T9 plotting board, 19 
T15 antiaircraft, 8, 14-15, 135, 
142 

T15-E1 curved flight, 133 
T28, intermediate range, 8, 30 

Gunsights, computing 
antitank, 32 

course-invariant, 8-9, 28-29 
gyroscopic lead-computing, 9, 44- 
46 

Mark 15; 45-46 
Mark 15-P, 46 

Impulsive admittance of linear net- 
works 

derivation, 112-114 
minimization of noise, 107 
series relationships, 152 
symmetry, 151-152 
theory, 145-147 
transmission function, 148 
variable transmission, 154-155 

Interpolation relay for computa- 
tion, 57 

Iowa State College, prediction de- 
vices, 20 

Jordan’s lemma, linear network 
theory, 150 

Land-based fire control 
see Fire control 

Laplace transform, linear network, 
148-150 

Borel’s theorem, 150 
Jordan’s lemma, 150 
translation theorem, 149-150 

Lawrence Aeronautical Corpora- 
tion, pneumatic techniques 
for fire control, 49-51 

Lead-computing sights, gyroscopic, 
9, 44-46 

Least squares criterion, data 
smoothing, 78-79, 104 

Leeds and Northrup Co., speed gen- 
erator, 41, 42 

Legendre polynomials, data smooth- 
ing, 115-116 

Librascope Corp., gun director 
Mark 56; 68-69 

Linear network theory, data 
smoothing 

see Network theory, data smooth- 
ing 


M3B1 oil gear power drive, 6 
M5A2 range director, intermediate, 
25 

M7 gun director (Weissight), 6, 17, 
28 

M9 gun director, antiaircraft, 6, 8, 
13-14, 16, 125-131 
Mark VII radar, tracking errors, 
128 

Mark 15 lead computing sight, 45- 
46 

Mark 15-P lead-computing sight, 
46 

Mark 23 bombsight. Navy, 9, 46-49 
Mark 25 bombsight. Navy, 49 
Mark 49 gun director, 66 
Mark 56 gun director, 8, 66-69 
Massachusetts Institute of Tech- 
nology, hydraulic remote- 
control servos, 40-41 
Mathematical analysis, fire-control 
problems, 54-61 
air warfare analysis, 61 
computations, 60 
mechanical aids, 56-59 
miscellaneous studies, 59-60 
statistical theory of prediction, 
54-56 

train bombing, statistics, 60 
McMath - Hulbert Observatory, 
gyroscopic lead-computing 
sights, 45, 47-48 

Mean square error, data smoothing, 
79, 104 


Navy bombsights 
Mark 23; 9, 46-49 
Mark 25; 9, 49 
Navy fire control, radar, 62-69 
Mark 49 gun director, 66 
Mark 56 gunfire-control system, 
66-69 

SCR-547 radar, 62-63 
seaborne torpedo projectors, 65- 
66 

T4 chronograph, 63-65 
Navy gyroscopic lead-computing 
gunsight Mark 15; 45-46 
Network theory, data smoothing, 
134-154 

impulsive admittance, 145-147 
Laplace transforms, 148-150 
polynomial solution, 138-139 
transmission function, 147-148 
variation, limited range, 139-141 
variation with target position, 
136-138 

variation with time, 138 



INDEX 


179 


Noise spectra, data smoothing, 86- 
87 

elementary pulse method, 157- 
159 

Phillips and Weiss theory, 156- 
157 

smoothing functions, 112, 157 
weighting functions, 107 

Observational error function, auto- 
correlation, 78-79 
Oil-gear power drive M3B1; 6 

Patuxent River testing project, 6, 
8 

Phase relations, target paths, 100- 
106 

Phillips and Weiss, t-method for 
smoothing functions, 118, 
156-157 

Pilot’s universal sighting system 
(PUSS), 8, 51 

Plane-to-plane fire control, 6-8, 61 
p-Methods, physical realization of 
weighting functions, 119-120 
Pneumatic fire-control techniques, 
9, 44-53 

angular rate indicator, 47 
BARB III bombsight, 46-48 
elements for torpedo control, 50 
gyroscopic lead-computing sights, 
44-46 

gyroscopic substitutes, 52-53 
Mark 15 gunsight, 45, 46 
Mark 15-P gunsight, 45 
Mark 23 bombsight, 46-49 
Mark 25 bombsight, 49 
Pilot’s universal sighting system 
(PUSS), 51 

stabilization of aerial cameras, 
51 

theory, 51-53 

vacuum regulator for Mark 25 
bombsight, 50 

Poisson distribution of segment 
end points, 101-102, 104-105 
Polaroid Corporatfon, range finders, 
32-33 

Polynomials, smoothing functions 
see Smoothing functions 
Position data smoothing, 131-133 
Prediction theory, fire-control de- 
vices, 85 

extrapolation, 56, 75-76, 112 
geometrical method, 19-20, 54 
network design, 97-98 
tracking errors, 127-130 
Wiener’s method, 55-56, 93-97 
PUSS (Pilot’s universal sighting 
system) , 8, 51 


Radar fire control. Navy, 62-69 
Mark 49 gun director, 66 
Mark 56 gunfire-control system, 
66-69 

SCR-547 radar, 62-63 
seaborne torpedo projectors, 65- 
66 

T4 chronograph, 63-65 
“Random noise” functions, data 
smoothing, 87-89 
Range directors 
see Gun directors 
Range finders, 22-23 
optical, 9-10 
Polaroid, 32-33 
radar, 62-63 
RC networks, 121-124 
RCA electronic computers, 20-21 
Recommendations for fire-control 
research, 6-10 
human factors, 9 
principles, 8-9 
range finders, 9-10 
testing devices, 7-8 
theory, 7 

Resistance - capacitance networks, 
121-124 

SCR-547 radar, fire control, 62-63 
Seaborne fire control, radar 
see Radar fire control. Navy 
Servomechanisms, 9, 38-43, 120- 
121, 133 

booster systems, small guns, 39 
circuits, 133 
clutch-type, 39 
guns, medium caliber, 39-41 
remote control, 40 
transmission systems, sea coast 
data, 42 

Sights, computing 
see Computing sights 
Smoothing circuits, variable and 
nonlinear, 134-143 
description, 141-143 
evaluation, 134, 142-143 
feedback, 134-135 
polynomial expansion, 138-139 
target position, 142-143 
three-dimensional, 135-136 
Smoothing functions, 107-116, 156- 
159 

autocorrelation method, 109-110 
curve-fitting method, 108-109 
elementary pulse method, 110-111, 
157-159 

exponential, 107-108 
impulsive admittance character- 
istics, 107 

Legendre polynomials, 115-116 



Phillips and Weiss theory, 156- 
157 

polynomials, 112-116 
prediction, 112 
symmetry, 157 

T4 chronograph, radar, 63-65 
T9 gun director, plotting board, 19 
T15 gun director, antiaircraft, 8, 
14-15, 135, 142 
T15-E1 flight director, 133 
T28 range director, 8, 30 
T62 antitank computing sight, 32 
Target courses, analytic arcs as- 
sumption 

see Analytic arcs assumption, 
target courses 

Testing devices, antiaircraft fire 
control, 6-8, 33-37 
ballistic computer, 35 
data recorder, 35 
dynamic testers, 6, 7, 33-35 
Patuxent River project, plane-to- 
plane, 6, 8 
relay interpolator, 35 
Texas tester, plane-to-plane, 6, 8 
weapons, automatic, 36-37 
Texas tester, plane-to-plane fire 
control, 6, 8 

Time series analysis, data smooth- 
ing, 76-79, 81 
autocorrelation, 77-78 
filters, 81 

least square criterion, 78-79 
prediction circuit equation, 77 
t-Methods, weighting functions, 
117-119 

Torpedo directors, seaborne, 65-66 
Translation theorem, linear net- 
works, 149-150 

Transmission function, linear net- 
work 

feedback amplifiers, 120-121 
impulsive admittance, 148 
physical restrictions, 153-154 
position data, 132 
power series expansion, 119 
second-derivative circuit, 125-126, 
130-131 

theory, 147-148 

effect,” feedback amplifiers, 
121 

United Shoe Manufacturing Corp., 
hydraulic boosters, 39 
University of North Carolina, anti- 
aircraft test equipment, 36 
University of Pennsylvania, differ- 
ential analyzer, 58 


180 


INDEX 


Variable smoothing in an electrical 
circuit, 139, 142 

Waugh Equipment Co. aircraft 
turret smoothness tester, 43 
Weighting functions 
derivatives, 114-115 
elementary pulse method, 157- 
159 

feedback amplifiers, 120-121 


Phillips and Weiss theory, 156- 
157 

p-methods, 119-120 
polynomials, 114 

resistance-capacitance networks, 
121-124 

servomechanisms, 120-121 
smoothing, 108-110 
symmetry, 157 
t-methods, 117-119 
variable smoothing time, 139 


Weiss and Phillips, t-method for 
smoothing functions, 118, 
156-157 

Weissight (M7 gun director), 6, 28 

Westinghouse Electric and Manu- 
facturing Co. 
servos, hydraulic, 41 
chronographs, 63 

Wiener’s prediction theory, fire- 
control devices, 55-56, 93-97 


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